8月19日-报告3-尹万科-Finite types conditions for real smooth hypersurfaces.pdf

Finite type conditions for real smooth hypersurfaces Wanke Yin School of Mathematics and Statistics, Wuhan University Aug. 19th§Shanghai Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan1Univers / 31 Finite type conditions for real smooth hypersurfaces Wanke Yin School of Mathematics and Statistics, Wuhan University Aug. 19th§Shanghai Finite type conditions arise naturally during the study of weakly pseudoconvex hypersurfaces in Cn . Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan3Univers / 31 Finite type conditions arise naturally during the study of weakly pseudoconvex hypersurfaces in Cn . For strongly pseudoconvex hypersurfaces (and their associated domains), there are many wonderful theorems. For example, Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan3Univers / 31 Finite type conditions arise naturally during the study of weakly pseudoconvex hypersurfaces in Cn . For strongly pseudoconvex hypersurfaces (and their associated domains), there are many wonderful theorems. For example, 1 Cartan-Chern-Moser Theorem Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan3Univers / 31 Finite type conditions arise naturally during the study of weakly pseudoconvex hypersurfaces in Cn . For strongly pseudoconvex hypersurfaces (and their associated domains), there are many wonderful theorems. For example, 1 Cartan-Chern-Moser Theorem 2 Kohn’s Sub-Elliptic Estimates Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan3Univers / 31 Cartan-Chern-Moser Theorem Poincaré: The real hypersurfaces in C2 are generically not locally holomorphic equivalent. Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan4Univers / 31 Cartan-Chern-Moser Theorem Poincaré: The real hypersurfaces in C2 are generically not locally holomorphic equivalent.
In contrast, any hypersurface in Rn locally diffeomorphic to (Rn−1 , 0), 0 . Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan4Univers / 31 Cartan-Chern-Moser Theorem Poincaré: The real hypersurfaces in C2 are generically not locally holomorphic equivalent.
In contrast, any hypersurface in Rn locally diffeomorphic to (Rn−1 , 0), 0 . Cartan: The strongly pseudoconvex real analytic hypersurface near the origin in C2 possesses the following normal form: X v = |z|2 + akl (u)z k z l . k,l≥2,k+l≥6 Here (z, w = u + iv) are the coordinates of C2 . Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan4Univers / 31 Cartan-Chern-Moser Theorem Poincaré: The real hypersurfaces in C2 are generically not locally holomorphic equivalent.
In contrast, any hypersurface in Rn locally diffeomorphic to (Rn−1 , 0), 0 . Cartan: The strongly pseudoconvex real analytic hypersurface near the origin in C2 possesses the following normal form: X v = |z|2 + akl (u)z k z l . k,l≥2,k+l≥6 Here (z, w = u + iv) are the coordinates of C2 . Chern-Moser: Normal form for real hypersurfaces in Cn . Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan4Univers / 31 This theorem gave a local classification of the real hypersurfaces up to the group SU (n, 1). Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan5Univers / 31 This theorem gave a local classification of the real hypersurfaces up to the group SU (n, 1). The pseudoconvex domain in Cn lack of the global invariants. The boundary invariants reflects the analytic and geometric properties of the associated domain. Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan5Univers / 31 This theorem gave a local classification of the real hypersurfaces up to the group SU (n, 1). The pseudoconvex domain in Cn lack of the global invariants. The boundary invariants reflects the analytic and geometric properties of the associated domain. Fefferman-Bochner: Two bounded smooth strongly pseudoconvex domains in Cn are biholomorphically equivalent if and only if their associated boundary are CR equivalence. Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan5Univers / 31 This theorem gave a local classification of the real hypersurfaces up to the group SU (n, 1). The pseudoconvex domain in Cn lack of the global invariants. The boundary invariants reflects the analytic and geometric properties of the associated domain. Fefferman-Bochner: Two bounded smooth strongly pseudoconvex domains in Cn are biholomorphically equivalent if and only if their associated boundary are CR equivalence. A natural question: What’s the local holomorphic invariants for general pseudoconvex hypersurfaces? Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan5Univers / 31 Kohn’s sub-elliptic estimates for strongly pseudoconvex domains(1963): Let D be a strongly pseudoconvex domain. Then we have the subelliptic estimates: Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan6Univers / 31 Kohn’s sub-elliptic estimates for strongly pseudoconvex domains(1963): Let D be a strongly pseudoconvex domain. Then we have the subelliptic estimates: ∗ For f ∈ Dom(∂) ∩ Dom(∂ ). Then 1 ∗ kf k2 ≤ k∂f k2 + k∂ f k2 + kf k2 with  = . 2 Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan6Univers / 31 Kohn’s sub-elliptic estimates for strongly pseudoconvex domains(1963): Let D be a strongly pseudoconvex domain. Then we have the subelliptic estimates: ∗ For f ∈ Dom(∂) ∩ Dom(∂ ). Then 1 ∗ kf k2 ≤ k∂f k2 + k∂ f k2 + kf k2 with  = . 2 This theorem is crucial on the investigation of the strongly pseudoconvex domains. Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan6Univers / 31 Kohn’s sub-elliptic estimates for strongly pseudoconvex domains(1963): Let D be a strongly pseudoconvex domain. Then we have the subelliptic estimates: ∗ For f ∈ Dom(∂) ∩ Dom(∂ ). Then 1 ∗ kf k2 ≤ k∂f k2 + k∂ f k2 + kf k2 with  = . 2 This theorem is crucial on the investigation of the strongly pseudoconvex domains. A natural question: Do the sub-elliptic estimates hold for general pseudoconvex domains? Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan6Univers / 31 Kohn showed that the sub-elliptic estimates does not always hold for general pseudoconvex domains. If D is a domain defined by n o 2 2 2 −1 r < 0, r(z1 , z2 , w) = Re(w) + |z12 + z23 |2 + exp−(|z1 | +|z2 | +|w| ) . Then there is no subelliptic estimate for (0, 1) forms near 0. Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan7Univers / 31 Kohn showed that the sub-elliptic estimates does not always hold for general pseudoconvex domains. If D is a domain defined by n o 2 2 2 −1 r < 0, r(z1 , z2 , w) = Re(w) + |z12 + z23 |2 + exp−(|z1 | +|z2 | +|w| ) . Then there is no subelliptic estimate for (0, 1) forms near 0. A natural question: What kind of pseudoconvex domains possess subelliptic estimates ? Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan7Univers / 31 J. Kohn (1972): Suppose M ⊂ C2 , p ∈ M . L: a (1, 0) tangential vector field near p with L(p) 6= 0. Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan8Univers / 31 J. Kohn (1972): Suppose M ⊂ C2 , p ∈ M . L: a (1, 0) tangential vector field near p with L(p) 6= 0. η: the purely imaginary non-vanishing 1 form, that annihilates the holomorphic vector bundle. Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan8Univers / 31 J. Kohn (1972): Suppose M ⊂ C2 , p ∈ M . L: a (1, 0) tangential vector field near p with L(p) 6= 0. η: the purely imaginary non-vanishing 1 form, that annihilates the holomorphic vector bundle. T : the totally imaginary tangent vector field such that η(T ) = 1. Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan8Univers / 31 J. Kohn (1972): Suppose M ⊂ C2 , p ∈ M . L: a (1, 0) tangential vector field near p with L(p) 6= 0. η: the purely imaginary non-vanishing 1 form, that annihilates the holomorphic vector bundle. T : the totally imaginary tangent vector field such that η(T ) = 1. We have the following invariants: Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan8Univers / 31 J. Kohn (1972): Suppose M ⊂ C2 , p ∈ M . L: a (1, 0) tangential vector field near p with L(p) 6= 0. η: the purely imaginary non-vanishing 1 form, that annihilates the holomorphic vector bundle. T : the totally imaginary tangent vector field such that η(T ) = 1. We have the following invariants: (1) Contact order by regular holomorphic curves a(1) (M, p), Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan8Univers / 31 J. Kohn (1972): Suppose M ⊂ C2 , p ∈ M . L: a (1, 0) tangential vector field near p with L(p) 6= 0. η: the purely imaginary non-vanishing 1 form, that annihilates the holomorphic vector bundle. T : the totally imaginary tangent vector field such that η(T ) = 1. We have the following invariants: (1) Contact order by regular holomorphic curves a(1) (M, p),  a(1) (M, p) = sup r| ∃ a local regular holomorphic curve γ γ whose order of vanishing of ρ|γ at p is r . Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan8Univers / 31 J. Kohn (1972): Suppose M ⊂ C2 , p ∈ M . L: a (1, 0) tangential vector field near p with L(p) 6= 0. η: the purely imaginary non-vanishing 1 form, that annihilates the holomorphic vector bundle. T : the totally imaginary tangent vector field such that η(T ) = 1. We have the following invariants: (1) Contact order by regular holomorphic curves a(1) (M, p),  a(1) (M, p) = sup r| ∃ a local regular holomorphic curve γ γ whose order of vanishing of ρ|γ at p is r . Here and in what follows, ρ is the defining function of M near p. Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan8Univers / 31 (2) Iterated Lie brackets t(1) (M, p), Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan9Univers / 31 (2) Iterated Lie brackets t(1) (M, p), Define t(L, p) = m if for any k ≤ m − 1 and L1 , · · · , Lk = L or L, we have Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan9Univers / 31 (2) Iterated Lie brackets t(1) (M, p), Define t(L, p) = m if for any k ≤ m − 1 and L1 , · · · , Lk = L or L, we have hη, [· · · [[L1 , L2 ], L3 ] · · · , Lk ]i(p) = 0. Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan9Univers / 31 (2) Iterated Lie brackets t(1) (M, p), Define t(L, p) = m if for any k ≤ m − 1 and L1 , · · · , Lk = L or L, we have hη, [· · · [[L1 , L2 ], L3 ] · · · , Lk ]i(p) = 0. But for a certain L1 , · · · , Lm = L or L, we have Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan9Univers / 31 (2) Iterated Lie brackets t(1) (M, p), Define t(L, p) = m if for any k ≤ m − 1 and L1 , · · · , Lk = L or L, we have hη, [· · · [[L1 , L2 ], L3 ] · · · , Lk ]i(p) = 0. But for a certain L1 , · · · , Lm = L or L, we have hη, [· · · [[L1 , L2 ], L3 ] · · · , Lm ]i(p) 6= 0. Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan9Univers / 31 (2) Iterated Lie brackets t(1) (M, p), Define t(L, p) = m if for any k ≤ m − 1 and L1 , · · · , Lk = L or L, we have hη, [· · · [[L1 , L2 ], L3 ] · · · , Lk ]i(p) = 0. But for a certain L1 , · · · , Lm = L or L, we have hη, [· · · [[L1 , L2 ], L3 ] · · · , Lm ]i(p) 6= 0. Then t(L, p) is independent of L and define t(1) (M, p) = t(L, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. and 19th§Shanghai Statistics, Wuhan9Univers / 31 (3) The degeneracy of the Levi form c(1) (M, p) Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 10Univers / 31 (3) The degeneracy of the Levi form c(1) (M, p) Define c(L, p) = m if for any k (≤ m − 3) vector fields L1 , · · · , Lk = L or L, it holds that Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 10Univers / 31 (3) The degeneracy of the Levi form c(1) (M, p) Define c(L, p) = m if for any k (≤ m − 3) vector fields L1 , · · · , Lk = L or L, it holds that L1 · · · Lk hη, [L, L]i(p) = 0 Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 10Univers / 31 (3) The degeneracy of the Levi form c(1) (M, p) Define c(L, p) = m if for any k (≤ m − 3) vector fields L1 , · · · , Lk = L or L, it holds that L1 · · · Lk hη, [L, L]i(p) = 0 Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 10Univers / 31 (3) The degeneracy of the Levi form c(1) (M, p) Define c(L, p) = m if for any k (≤ m − 3) vector fields L1 , · · · , Lk = L or L, it holds that L1 · · · Lk hη, [L, L]i(p) = 0 and for a certain choice of m − 2 vector fields L1 , · · · , Lm−2 = L or L, it holds that Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 10Univers / 31 (3) The degeneracy of the Levi form c(1) (M, p) Define c(L, p) = m if for any k (≤ m − 3) vector fields L1 , · · · , Lk = L or L, it holds that L1 · · · Lk hη, [L, L]i(p) = 0 and for a certain choice of m − 2 vector fields L1 , · · · , Lm−2 = L or L, it holds that L1 · · · Lm−2 hη, [L, L]i(p) 6= 0. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 10Univers / 31 (3) The degeneracy of the Levi form c(1) (M, p) Define c(L, p) = m if for any k (≤ m − 3) vector fields L1 , · · · , Lk = L or L, it holds that L1 · · · Lk hη, [L, L]i(p) = 0 and for a certain choice of m − 2 vector fields L1 , · · · , Lm−2 = L or L, it holds that L1 · · · Lm−2 hη, [L, L]i(p) 6= 0. Then c(L, p) is independent of L and define c(1) (M, p) = c(L, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 10Univers / 31 (4) Contact order by holomorphic curves ∆1 (M, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 11Univers / 31 (4) Contact order by holomorphic curves ∆1 (M, p). ∆1 (M, 0) = Wanke Yin µ(z ∗ r) . z:(C,0)→(Cn ,z0 ) µ(z) sup Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 11Univers / 31 (4) Contact order by holomorphic curves ∆1 (M, p). ∆1 (M, 0) = µ(z ∗ r) . z:(C,0)→(Cn ,z0 ) µ(z) sup Here z is a local holomorphic curve near 0. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 11Univers / 31 (4) Contact order by holomorphic curves ∆1 (M, p). ∆1 (M, 0) = µ(z ∗ r) . z:(C,0)→(Cn ,z0 ) µ(z) sup Here z is a local holomorphic curve near 0. When z is required to be regular, this is exactly the regular finite type a(1) (M, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 11Univers / 31 J. Kohn (1972): Theorem: a(1) (M, p) = t(1) (M, p) = c(1) (M, p) = ∆1 (M, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 12Univers / 31 J. Kohn (1972): Theorem: a(1) (M, p) = t(1) (M, p) = c(1) (M, p) = ∆1 (M, p). pseudoconvexity is not necessary in the theorem. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 12Univers / 31 J. Kohn (1972): Theorem: a(1) (M, p) = t(1) (M, p) = c(1) (M, p) = ∆1 (M, p). pseudoconvexity is not necessary in the theorem. When M is pseudoconvex, these invariants = m if and only if 1 (1) subelliptic estimates holds for  = m , but (2) (Greiner 1974) for no large value of . Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 12Univers / 31 J. Kohn (1972): Theorem: a(1) (M, p) = t(1) (M, p) = c(1) (M, p) = ∆1 (M, p). pseudoconvexity is not necessary in the theorem. When M is pseudoconvex, these invariants = m if and only if 1 (1) subelliptic estimates holds for  = m , but (2) (Greiner 1974) for no large value of . This finite type at 0 ∈ M is of finite type m if and only if the defining function can take the following form ρ = Im(w) + P (z, z) + O(|z|m+1 + |zRe(w)| + |Re(w)|2 ). Here P is a non trivial homogeneous polynomial of degree m without harmonic terms. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 12Univers / 31 Generalization of Kohn’s notion of the boundary finite type condition to higher dimensions has been a subject under extensive investigations in the past 40 years in Several Complex Variables. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 13Univers / 31 Generalization of Kohn’s notion of the boundary finite type condition to higher dimensions has been a subject under extensive investigations in the past 40 years in Several Complex Variables. Different measurements of the degeneracy of the Levi form results in the different finite types. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 13Univers / 31 Generalization of Kohn’s notion of the boundary finite type condition to higher dimensions has been a subject under extensive investigations in the past 40 years in Several Complex Variables. Different measurements of the degeneracy of the Levi form results in the different finite types. regular finite type (of Bloom-Graham) Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 13Univers / 31 Generalization of Kohn’s notion of the boundary finite type condition to higher dimensions has been a subject under extensive investigations in the past 40 years in Several Complex Variables. Different measurements of the degeneracy of the Levi form results in the different finite types. regular finite type (of Bloom-Graham) D’Angelo finite type Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 13Univers / 31 Generalization of Kohn’s notion of the boundary finite type condition to higher dimensions has been a subject under extensive investigations in the past 40 years in Several Complex Variables. Different measurements of the degeneracy of the Levi form results in the different finite types. regular finite type (of Bloom-Graham) D’Angelo finite type Catlin multitype Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 13Univers / 31 Generalization of Kohn’s notion of the boundary finite type condition to higher dimensions has been a subject under extensive investigations in the past 40 years in Several Complex Variables. Different measurements of the degeneracy of the Levi form results in the different finite types. regular finite type (of Bloom-Graham) D’Angelo finite type Catlin multitype Catlin finite type Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 13Univers / 31 T. Bloom (1981): When M ⊂ Cn . For each integer 1 ≤ s ≤ n − 1, we can similarly define corresponding integer invariants: Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 14Univers / 31 T. Bloom (1981): When M ⊂ Cn . For each integer 1 ≤ s ≤ n − 1, we can similarly define corresponding integer invariants: 1 The s-contact type a(s) (M, p) Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 14Univers / 31 T. Bloom (1981): When M ⊂ Cn . For each integer 1 ≤ s ≤ n − 1, we can similarly define corresponding integer invariants: 1 The s-contact type a(s) (M, p) 2 The s-vector field type t(s) (M, p), Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 14Univers / 31 T. Bloom (1981): When M ⊂ Cn . For each integer 1 ≤ s ≤ n − 1, we can similarly define corresponding integer invariants: 1 The s-contact type a(s) (M, p) 2 The s-vector field type t(s) (M, p), 3 The s-type of the Levi form c(s) (M, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 14Univers / 31 The first invariant is more of algebraic, comparatively more easily to compute Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 15Univers / 31 The first invariant is more of algebraic, comparatively more easily to compute The second is defined in a way more of differential geometry Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 15Univers / 31 The first invariant is more of algebraic, comparatively more easily to compute The second is defined in a way more of differential geometry The third invariant is defined by the degeneracy of the Levi form, it is always more easily to be applied. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 15Univers / 31 Bloom-Graham (1977): a(n−1) (M, p) = t(n−1) (M, p). Bloom (1978): a(n−1) (M, p) = c(n−1) (M, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 16Univers / 31 Bloom-Graham (1977): a(n−1) (M, p) = t(n−1) (M, p). Bloom (1978): a(n−1) (M, p) = c(n−1) (M, p). Bloom (1981): For any 1 ≤ s ≤ n − 1, a(s) (M, p) ≤ t(s) (M, p), a(s) (M, p) ≤ c(s) (M, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 16Univers / 31 Bloom-Graham (1977): a(n−1) (M, p) = t(n−1) (M, p). Bloom (1978): a(n−1) (M, p) = c(n−1) (M, p). Bloom (1981): For any 1 ≤ s ≤ n − 1, a(s) (M, p) ≤ t(s) (M, p), a(s) (M, p) ≤ c(s) (M, p). For these results, pseudo-convexity is not necessary. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 16Univers / 31 T. Bloom 1981 Conjecture: When M is pseudo-convex, for 1 ≤ s ≤ n−1, a(s) (M, p) = t(s) (M, p) = c(s) (M, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 17Univers / 31 T. Bloom 1981 Conjecture: When M is pseudo-convex, for 1 ≤ s ≤ n−1, a(s) (M, p) = t(s) (M, p) = c(s) (M, p). pseudo-convexity is necessary in this conjecture: Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 17Univers / 31 T. Bloom 1981 Conjecture: When M is pseudo-convex, for 1 ≤ s ≤ n−1, a(s) (M, p) = t(s) (M, p) = c(s) (M, p). pseudo-convexity is necessary in this conjecture: Let ρ = 2Re(w) + (z2 + z2 + |z1 |2 )2 and let M = {(z1 , z2 , w) ∈ C3 | ρ = 0}. Let p = (0, 0, 0). Then a(1) (M, p) = 4 but c(1) (M, p) = t(1) (M, p) = ∞ . Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 17Univers / 31 T. Bloom 1981 Conjecture: When M is pseudo-convex, for 1 ≤ s ≤ n−1, a(s) (M, p) = t(s) (M, p) = c(s) (M, p). pseudo-convexity is necessary in this conjecture: Let ρ = 2Re(w) + (z2 + z2 + |z1 |2 )2 and let M = {(z1 , z2 , w) ∈ C3 | ρ = 0}. Let p = (0, 0, 0). Then a(1) (M, p) = 4 but c(1) (M, p) = t(1) (M, p) = ∞ . When M ⊂ C3 , a(1) (M, p) = c(1) (M, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 17Univers / 31 Huang-Y. (2021): When M is pseudo-convex, a(n−2) (M, p) = t(n−2) (M, p) = c(n−2) (M, p). In particular, this gives a complete solution for n = 3. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 18Univers / 31 Huang-Y. (2021): When M is pseudo-convex, a(n−2) (M, p) = t(n−2) (M, p) = c(n−2) (M, p). In particular, this gives a complete solution for n = 3. Chen-Chen-Y. (2021): Suppose that M is pseudo-convex, the Levi form at p has only one degenerate eigenvalue. Then a(1) (M, p) = t(1) (M, p) = c(1) (M, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 18Univers / 31 Huang-Y. (2021): When M is pseudo-convex, a(n−2) (M, p) = t(n−2) (M, p) = c(n−2) (M, p). In particular, this gives a complete solution for n = 3. Chen-Chen-Y. (2021): Suppose that M is pseudo-convex, the Levi form at p has only one degenerate eigenvalue. Then a(1) (M, p) = t(1) (M, p) = c(1) (M, p). (In this case, a(1) (M, p) = c(1) (M, p) is due to Abdallah TALHAOUI (1983)) Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 18Univers / 31 A Conjecture of D’Angelo (1986) For a fixed tangent (1, 0) vector field L, as in C2 , we can similarly define t(L, p) and c(L, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 19Univers / 31 A Conjecture of D’Angelo (1986) For a fixed tangent (1, 0) vector field L, as in C2 , we can similarly define t(L, p) and c(L, p). For higher dimensional case, t(L, p) and c(L, p) depends on L. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 19Univers / 31 A Conjecture of D’Angelo (1986) For a fixed tangent (1, 0) vector field L, as in C2 , we can similarly define t(L, p) and c(L, p). For higher dimensional case, t(L, p) and c(L, p) depends on L. D’Angelo Conjecture: Let M be a pseudoconvex smooth hypersurface, p ∈ M . Then for any fixed (1, 0) tangent vector field L, we have t(L, p) = c(L, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 19Univers / 31 A Conjecture of D’Angelo (1986) For a fixed tangent (1, 0) vector field L, as in C2 , we can similarly define t(L, p) and c(L, p). For higher dimensional case, t(L, p) and c(L, p) depends on L. D’Angelo Conjecture: Let M be a pseudoconvex smooth hypersurface, p ∈ M . Then for any fixed (1, 0) tangent vector field L, we have t(L, p) = c(L, p). It implies one equality of the Bloom Conjecture. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 19Univers / 31 Progress on the D’Angelo Conjecture D’Angelo 1986: t(L, p) = 4 if and only if c(L, p) = 4. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 20Univers / 31 Progress on the D’Angelo Conjecture D’Angelo 1986: t(L, p) = 4 if and only if c(L, p) = 4. Chen-Y.-Yuan 2020: t(L, p) = c(L, p) if n = 3. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 20Univers / 31 Progress on the D’Angelo Conjecture D’Angelo 1986: t(L, p) = 4 if and only if c(L, p) = 4. Chen-Y.-Yuan 2020: t(L, p) = c(L, p) if n = 3. Chen-Chen-Y. (2021): Suppose that M is pseudo-convex, the Levi form at p has only one degenerate eigenvalue. Then, for any fixed (1, 0) tangent vector field L, we have t(L, p) = c(L, p). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 20Univers / 31 Progress on the D’Angelo Conjecture D’Angelo 1986: t(L, p) = 4 if and only if c(L, p) = 4. Chen-Y.-Yuan 2020: t(L, p) = c(L, p) if n = 3. Chen-Chen-Y. (2021): Suppose that M is pseudo-convex, the Levi form at p has only one degenerate eigenvalue. Then, for any fixed (1, 0) tangent vector field L, we have t(L, p) = c(L, p). Fassina (2018) tried to prove t(L, 0) ≥ c(L, 0). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 20Univers / 31 Progress on the D’Angelo Conjecture D’Angelo 1986: t(L, p) = 4 if and only if c(L, p) = 4. Chen-Y.-Yuan 2020: t(L, p) = c(L, p) if n = 3. Chen-Chen-Y. (2021): Suppose that M is pseudo-convex, the Levi form at p has only one degenerate eigenvalue. Then, for any fixed (1, 0) tangent vector field L, we have t(L, p) = c(L, p). Fassina (2018) tried to prove t(L, 0) ≥ c(L, 0). Recently, we have made some new progress on this problem. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 20Univers / 31 Kohn’s case (n=2) WLOG, we assume p = 0. In C2 case, for any two (1, 0) tangent vectors L and L0 with L(0), L0 (0) 6= 0, we have L = f L0 with f (0) 6= 0. Hence t(L, 0) = t(1) (M, 0), c(L, 0) = c(1) (M, 0). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 21Univers / 31 Kohn’s case (n=2) WLOG, we assume p = 0. In C2 case, for any two (1, 0) tangent vectors L and L0 with L(0), L0 (0) 6= 0, we have L = f L0 with f (0) 6= 0. Hence t(L, 0) = t(1) (M, 0), c(L, 0) = c(1) (M, 0). The first approach is to achieve the equality via a(1) (M, 0). Namely, we prove t(L, 0) = a(1) (M, 0), c(L, 0) = a(1) (M, 0). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 21Univers / 31 Kohn’s case (n=2) In nornal form, We may assume ρ = Im(w) + P (z, z) + O(|z|m+1 + |zRe(w)| + |Re(w)|2 ). Then t(L, 0) and c(L, 0) can be obtained by direct computation. In fact, if L= Wanke Yin ∂ ∂ − ρz (ρw )−1 . ∂z ∂w Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 22Univers / 31 Kohn’s case (n=2) In nornal form, We may assume ρ = Im(w) + P (z, z) + O(|z|m+1 + |zRe(w)| + |Re(w)|2 ). Then t(L, 0) and c(L, 0) can be obtained by direct computation. In fact, if L= ∂ ∂ − ρz (ρw )−1 . ∂z ∂w Then we have the following explicit formulas: o 1  2 2 λ(L, L) = ρzz |ρw | − 2Re(ρzw ρw ρz ) + rww |ρz | . |ρw |2 [∂ρ, [· · · [[L, L], L1 ], · · · , Lm−2 ](0) = ∂ r+s ρ(0). ∂z r ∂z s Here L1 , · · · , Lm−2 = L or L, r and s are the numbers of L and L. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 22Univers / 31 Higher dimensional case For Bloom-Graham’s case, the proofs of a(n−1) (M, 0) = t(n−1) (M, 0) and a(n−1) (M, 0) = c(n−1) (M, 0) are more or less the same. We still have the normal form and explicit computation. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 23Univers / 31 Higher dimensional case For Bloom-Graham’s case, the proofs of a(n−1) (M, 0) = t(n−1) (M, 0) and a(n−1) (M, 0) = c(n−1) (M, 0) are more or less the same. We still have the normal form and explicit computation. In the above cases, the pseudoconvex is not needed. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 23Univers / 31 Higher dimensional case For Bloom-Graham’s case, the proofs of a(n−1) (M, 0) = t(n−1) (M, 0) and a(n−1) (M, 0) = c(n−1) (M, 0) are more or less the same. We still have the normal form and explicit computation. In the above cases, the pseudoconvex is not needed. When we deal with the Bloom Conjecture and the D’Angelo Conjecture in higher dimensional case, the pseudoconvex is necessary. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 23Univers / 31 Higher dimensional case For Bloom-Graham’s case, the proofs of a(n−1) (M, 0) = t(n−1) (M, 0) and a(n−1) (M, 0) = c(n−1) (M, 0) are more or less the same. We still have the normal form and explicit computation. In the above cases, the pseudoconvex is not needed. When we deal with the Bloom Conjecture and the D’Angelo Conjecture in higher dimensional case, the pseudoconvex is necessary. The difficulty of these problems lies in how to make use this pseudoconvex condition. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 23Univers / 31 As for the C3 case, to prove the D’Angelo Conjecture, we made use of the Bloom Conjecture. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 24Univers / 31 As for the C3 case, to prove the D’Angelo Conjecture, we made use of the Bloom Conjecture. Let L be a general tangent vector field, which does not achieve the maximum for the commutator type or the Levi form type. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 24Univers / 31 As for the C3 case, to prove the D’Angelo Conjecture, we made use of the Bloom Conjecture. Let L be a general tangent vector field, which does not achieve the maximum for the commutator type or the Levi form type. We constructed a new hypersurface M 0 , and use the Bloom Conjecture to get a(1) (M 0 , 0) = t(L, 0) and a(1) (M 0 , 0) = c(L, 0). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 24Univers / 31 As for the C3 case, to prove the D’Angelo Conjecture, we made use of the Bloom Conjecture. Let L be a general tangent vector field, which does not achieve the maximum for the commutator type or the Levi form type. We constructed a new hypersurface M 0 , and use the Bloom Conjecture to get a(1) (M 0 , 0) = t(L, 0) and a(1) (M 0 , 0) = c(L, 0). For higher dimensional case (n ≥ 4), the Bloom Conjecture itself is still unknown. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 24Univers / 31 The other approach to prove the D’Angelo Conjecture is to obtain a direct relation between t(L, 0) and c(L, 0). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 25Univers / 31 The other approach to prove the D’Angelo Conjecture is to obtain a direct relation between t(L, 0) and c(L, 0). For L1 , · · · , Lk+1 = L or L and any tangent vector field L0 , define αL0 = η([T, L0 ]), and Γk+2 = [· · · [[L, L], L1 ] · · · , Lk ]. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 25Univers / 31 If Lk = L, then η(Γk+2 ) = η([Γk+1 , L]) = (αL − L)η(Γk+1 ) − λ(L, π0,1 Γk+1 ). If Lk = L, then η(Γk+2 ) = η([Γk+1 , L]) = (αL − L)η(Γk+1 ) − λ(π1,0 Γk+1 , L). The crucial fact in C2 is that there always exists a function f such that π1,0 Γk+1 = f L, π0,1 Γk+1 = gL. Wanke Yin (∗) Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 26Univers / 31 If Lk = L, then η(Γk+2 ) = η([Γk+1 , L]) = (αL − L)η(Γk+1 ) − λ(L, π0,1 Γk+1 ). If Lk = L, then η(Γk+2 ) = η([Γk+1 , L]) = (αL − L)η(Γk+1 ) − λ(π1,0 Γk+1 , L). The crucial fact in C2 is that there always exists a function f such that π1,0 Γk+1 = f L, π0,1 Γk+1 = gL. (∗) Thus by induction, η(Γk+2 ) = k Y (αLj − Lj )λ(L, L) + Pk−1 λ(L, L). j=1 Pj is a differential operator of order at most j along L and L. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 26Univers / 31 (∗) is crucial for the C2 case. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 27Univers / 31 (∗) is crucial for the C2 case. It does not hold for higher dimensional case, which made the problem extremely difficult. For example, Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 27Univers / 31 (∗) is crucial for the C2 case. It does not hold for higher dimensional case, which made the problem extremely difficult. For example, 1 Are t(L, 0) and c(L, 0) always even? Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 27Univers / 31 (∗) is crucial for the C2 case. It does not hold for higher dimensional case, which made the problem extremely difficult. For example, 1 Are t(L, 0) and c(L, 0) always even? 2 Is vL (f ) ≥ vL (g) if 0 ≤ f ≤ g? Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 27Univers / 31 (∗) is crucial for the C2 case. It does not hold for higher dimensional case, which made the problem extremely difficult. For example, 1 Are t(L, 0) and c(L, 0) always even? 2 Is vL (f ) ≥ vL (g) if 0 ≤ f ≤ g? Here VL (f ) is the vanishing order of f along L and L. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 27Univers / 31 (∗) is crucial for the C2 case. It does not hold for higher dimensional case, which made the problem extremely difficult. For example, 1 Are t(L, 0) and c(L, 0) always even? 2 Is vL (f ) ≥ vL (g) if 0 ≤ f ≤ g? Here VL (f ) is the vanishing order of f along L and L. The second question is trivial if L is a real tangent vector field. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 27Univers / 31 Direct connection for higher dimensional case Write Lm+2 = [· · · [[L, L], L1 ] · · · , Lm ] Lj = L or L. Then if Lm = L η(Lm+2 ) = (αL − L)η(Lm+1 ) − λ(L, Π0,1 Lm+1 ) If if Lm = L η(Lm+2 ) = (αL − L)η(Lm+1 ) + λ(Π1,0 Lm+1 , L). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 28Univers / 31 Direct connection for higher dimensional case Write Lm+2 = [· · · [[L, L], L1 ] · · · , Lm ] Lj = L or L. Then if Lm = L η(Lm+2 ) = (αL − L)η(Lm+1 ) − λ(L, Π0,1 Lm+1 ) If if Lm = L η(Lm+2 ) = (αL − L)η(Lm+1 ) + λ(Π1,0 Lm+1 , L). Hence by induction, we obtain η(Lm+2 ) = (−1)m Lm · · · L1 λ(L, L) + R. R is extremely complicated, it is no longer lower times derivative of λ(L, L) along L and L. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 28Univers / 31 Write X = Π1,0 [L, L]. In the case of t = 4 or c = 4, η([[[L, L], L], L]) + η([[[L, L], L], L]) = (LL + LL)λ(L, L) + 2λ(X, X). Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 29Univers / 31 Write X = Π1,0 [L, L]. In the case of t = 4 or c = 4, η([[[L, L], L], L]) + η([[[L, L], L], L]) = (LL + LL)λ(L, L) + 2λ(X, X). The key point is that both (LL + LL)λ(L, L) and the remainder term λ(X, X) are positive, due to the pseudoconvexity. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 29Univers / 31 Write X = Π1,0 [L, L]. In the case of t = 4 or c = 4, η([[[L, L], L], L]) + η([[[L, L], L], L]) = (LL + LL)λ(L, L) + 2λ(X, X). The key point is that both (LL + LL)λ(L, L) and the remainder term λ(X, X) are positive, due to the pseudoconvexity. It is not easy to achieve such a positive remainder term even for the degree 6 case. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 29Univers / 31 Relation between these invariants Example: Let M ⊂ C4 be a real hypersurface defined by r = −2Imw + |z1 |4 + |z1 |2 |z2 |2 + |z1 |2 |z3 |2 + |z22 − z33 |4 . Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 30Univers / 31 Relation between these invariants Example: Let M ⊂ C4 be a real hypersurface defined by r = −2Imw + |z1 |4 + |z1 |2 |z2 |2 + |z1 |2 |z3 |2 + |z22 − z33 |4 . The Caltin multitypes at 0 are 4, 4, 4, The Bloom regular contact types are 4, 8, 12, The D’Angelo finite types are 4, 8, +∞. Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 30Univers / 31 Thank you for your attention! Wanke Yin Finite type conditions ( School of Mathematics Aug. 19th§Shanghai and Statistics, Wuhan 31Univers / 31