Variational Geometry
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1 Variational Geometry Hung Tran Texas Tech University Feb 20th, 2018 Junior Scholar Symposium, Texas Tech University Hung Tran (TTU) Variational Geometry Feb 20th, / 15
2 Geometric Variational Problems Hung Tran (TTU) Variational Geometry Feb 20th, / 15
3 Geometric Variational Problems Geometry: Metric (measurement) and curvature (shape). Hung Tran (TTU) Variational Geometry Feb 20th, / 15
4 Geometric Variational Problems Geometry: Metric (measurement) and curvature (shape). Ricci flow: Specific variations. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
5 Geometric Variational Problems Geometry: Metric (measurement) and curvature (shape). Ricci flow: Specific variations. Einstein Structures: Critical points of a natural geometric functional. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
6 Geometric Variational Problems Geometry: Metric (measurement) and curvature (shape). Ricci flow: Specific variations. Einstein Structures: Critical points of a natural geometric functional. Minimal surfaces: Critical points of the area functional. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
7 Background Differentiable Manifolds Hung Tran (TTU) Variational Geometry Feb 20th, / 15
8 Background Differentiable Manifolds Differentiable manifold M n : locally Euclidean. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
9 Background Differentiable Manifolds Differentiable manifold M n : locally Euclidean. Riemannian metric g: measure length/distance/volume. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
10 Background Curvature Hung Tran (TTU) Variational Geometry Feb 20th, / 15
11 Background Curvature Levi-Civita connection: Allow differentiation. Figure: A tangent plane Hung Tran (TTU) Variational Geometry Feb 20th, / 15
12 Background Curvature Levi-Civita connection: Allow differentiation. Figure: A tangent plane Curvature: (determined by derivatives of the metric) Measure non-flatness. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
13 Background Curvature Levi-Civita connection: Allow differentiation. Figure: A tangent plane Curvature: (determined by derivatives of the metric) Measure non-flatness. Intrinsic: Riemannian curvature, Ricci curvature, scalar curvature. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
14 Background Curvature Levi-Civita connection: Allow differentiation. Figure: A tangent plane Curvature: (determined by derivatives of the metric) Measure non-flatness. Intrinsic: Riemannian curvature, Ricci curvature, scalar curvature. Extrinsic: Mean curvature, second fundamental form. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
15 Background Space Forms Constant curvature models: Euclidean space (flat), round sphere (constant positive), hyperbolic space (constant negative). Hung Tran (TTU) Variational Geometry Feb 20th, / 15
16 Background Space Forms Constant curvature models: Euclidean space (flat), round sphere (constant positive), hyperbolic space (constant negative). Figure: Round sphere Hung Tran (TTU) Variational Geometry Feb 20th, / 15
17 Background Space Forms Constant curvature models: Euclidean space (flat), round sphere (constant positive), hyperbolic space (constant negative). Figure: Round sphere Figure: Poincare model Hung Tran (TTU) Variational Geometry Feb 20th, / 15
18 Background Space Forms Constant curvature models: Euclidean space (flat), round sphere (constant positive), hyperbolic space (constant negative). Figure: Round sphere Figure: Poincare model Hung Tran (TTU) Variational Geometry Feb 20th, / 15
19 Ricci Flow Ricci Flow 1 J. Hyam Rubinstein and Robert Sinclair. Visualizing Ricci Flow of Manifolds of Revolution, Experimental Mathematics v. 14 n. 3, pp Hung Tran (TTU) Variational Geometry Feb 20th, / 15
20 Ricci Flow Ricci Flow (M, g(t)) is a Ricci flow solution if t g = 2Rc. 1 J. Hyam Rubinstein and Robert Sinclair. Visualizing Ricci Flow of Manifolds of Revolution, Experimental Mathematics v. 14 n. 3, pp Hung Tran (TTU) Variational Geometry Feb 20th, / 15
21 Ricci Flow Ricci Flow (M, g(t)) is a Ricci flow solution if t g = 2Rc. Figure: Ricci flow on a neck 1 1 J. Hyam Rubinstein and Robert Sinclair. Visualizing Ricci Flow of Manifolds of Revolution, Experimental Mathematics v. 14 n. 3, pp Hung Tran (TTU) Variational Geometry Feb 20th, / 15
22 Ricci Flow Overview Hung Tran (TTU) Variational Geometry Feb 20th, / 15
23 Ricci Flow Overview Fundamental questions: Convergence. Formulation of Singularities. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
24 Ricci Flow Overview Fundamental questions: Convergence. Formulation of Singularities. Celebrated applications: G. Perelman s proof of the Poincare s conjecture. The proof of the differentiable sphere theorem by S. Brendle and R. Schoen. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
25 Ricci Flow Overview Fundamental questions: Convergence. Formulation of Singularities. Celebrated applications: G. Perelman s proof of the Poincare s conjecture. The proof of the differentiable sphere theorem by S. Brendle and R. Schoen. Technicality: Parabolic PDE, maximum principle. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
26 Ricci Flow Contributions Harnack inequalities crucial in Perelman s singularity analysis. Obtain analogous estimates in generalized settings: Ricci flow on warped Products (2015, JGA) (with Mihai Bailesteanu) Ricci-Harmonic map flow (2017, PEMS) (with Xiaodong Cao, Hongxin Guo) Generalized abstract flow (2015, MZ) (with X. Cao) Behavior of curvature towards the singular time (2015, MRL) Hung Tran (TTU) Variational Geometry Feb 20th, / 15
27 Einstein Structures Einstein Structures Hung Tran (TTU) Variational Geometry Feb 20th, / 15
28 Einstein Structures Einstein Structures (M, g) is an Einstein structure if, for a constant λ, Rc = λg. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
29 Einstein Structures Einstein Structures (M, g) is an Einstein structure if, for a constant λ, Rc = λg. Critical points of the Hilbert functional. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
30 Einstein Structures Einstein Structures (M, g) is an Einstein structure if, for a constant λ, Rc = λg. Critical points of the Hilbert functional. Generalized Structures: Gradient Ricci soliton, Harmonic curvature, Harmonic Weyl tensor. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
31 Einstein Structures Einstein Structures (M, g) is an Einstein structure if, for a constant λ, Rc = λg. Critical points of the Hilbert functional. Generalized Structures: Gradient Ricci soliton, Harmonic curvature, Harmonic Weyl tensor. Quest for the best metric. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
32 Einstein Structures Einstein Structures (M, g) is an Einstein structure if, for a constant λ, Rc = λg. Critical points of the Hilbert functional. Generalized Structures: Gradient Ricci soliton, Harmonic curvature, Harmonic Weyl tensor. Quest for the best metric. Figure: Round sphere Hung Tran (TTU) Variational Geometry Feb 20th, / 15
33 Einstein Structures Einstein Structures (M, g) is an Einstein structure if, for a constant λ, Rc = λg. Critical points of the Hilbert functional. Generalized Structures: Gradient Ricci soliton, Harmonic curvature, Harmonic Weyl tensor. Quest for the best metric. Figure: Round sphere Figure: Non-Round sphere Hung Tran (TTU) Variational Geometry Feb 20th, / 15
34 Einstein Structures Hung Tran (TTU) Variational Geometry Feb 20th, / 15
35 Einstein Structures Fundamental questions: Hung Tran (TTU) Variational Geometry Feb 20th, / 15
36 Einstein Structures Fundamental questions: Existence. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
37 Einstein Structures Fundamental questions: Existence. Uniqueness/moduli space. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
38 Einstein Structures Fundamental questions: Existence. Uniqueness/moduli space. Open question: Conjecture A non-flat simply connected Einstein four-manifold with non-negative sectional curvature must be either S 4, CP 2, S 2 S 2. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
39 Einstein Structures Fundamental questions: Existence. Uniqueness/moduli space. Open question: Conjecture A non-flat simply connected Einstein four-manifold with non-negative sectional curvature must be either S 4, CP 2, S 2 S 2. Technicality: Non-linear PDE, elliptic methods. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
40 Einstein Structures Contributions (with X. Cao) Rigidity of a gradient Ricci soliton (2016, GT) Rigidity of closed manifolds with harmonic Weyl curvature (2017, AiM) (with X. Cao) Progress towards E4M conjecture (2016, Preprint) Hung Tran (TTU) Variational Geometry Feb 20th, / 15
41 Minimal Surfaces Free Boundary Minimal Surfaces 2 Images courtesy of Peter McGrath Hung Tran (TTU) Variational Geometry Feb 20th, / 15
42 Minimal Surfaces Free Boundary Minimal Surfaces Σ B3, Σ B3, Σ is a FBMS if H 0 and Σ meets B3 perpendicularly. 2 Images courtesy of Peter McGrath Hung Tran (TTU) Variational Geometry Feb 20th, / 15
43 Minimal Surfaces Free Boundary Minimal Surfaces Σ B3, Σ B3, Σ is a FBMS if H 0 and Σ meets B3 perpendicularly. FBMS are critical points of the area functional with the free boundary condition (extension of Plateau s problem). Figure: Critical Catenoid 2 2 Images courtesy of Peter McGrath Hung Tran (TTU) Variational Geometry Feb 20th, / 15
44 Minimal Surfaces Overview Fundamental questions: Hung Tran (TTU) Variational Geometry Feb 20th, / 15
45 Minimal Surfaces Overview Fundamental questions: Regularity (relatively well understood). Hung Tran (TTU) Variational Geometry Feb 20th, / 15
46 Minimal Surfaces Overview Fundamental questions: Regularity (relatively well understood). Existence (rapid progress recently). Hung Tran (TTU) Variational Geometry Feb 20th, / 15
47 Minimal Surfaces Overview Fundamental questions: Regularity (relatively well understood). Existence (rapid progress recently). Uniqueness. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
48 Minimal Surfaces Overview Fundamental questions: Regularity (relatively well understood). Existence (rapid progress recently). Uniqueness. Analogous Lawson s conjecture: Conjecture A free boundary minimal annulus must be the critical catenoid. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
49 Minimal Surfaces Overview Fundamental questions: Regularity (relatively well understood). Existence (rapid progress recently). Uniqueness. Analogous Lawson s conjecture: Conjecture A free boundary minimal annulus must be the critical catenoid. Technicality: Elliptic PDE, PDE, GMT, and complex methods. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
50 Minimal Surfaces Contributions Hung Tran (TTU) Variational Geometry Feb 20th, / 15
51 Minimal Surfaces Contributions Stability (quantitatively measured by the Morse index) is crucial to answer uniqueness questions. Hung Tran (TTU) Variational Geometry Feb 20th, / 15
52 Minimal Surfaces Contributions Stability (quantitatively measured by the Morse index) is crucial to answer uniqueness questions. Develop a natural method to compute the Morse index (2016, CAG). Hung Tran (TTU) Variational Geometry Feb 20th, / 15
53 Minimal Surfaces Contributions Stability (quantitatively measured by the Morse index) is crucial to answer uniqueness questions. Develop a natural method to compute the Morse index (2016, CAG). (with Graham Smith, Ari Stern, and Detang Zhou) Study the growth of Morse indices of higher dimensional catenoids (2017, Preprint). Hung Tran (TTU) Variational Geometry Feb 20th, / 15
54 Minimal Surfaces Contributions Stability (quantitatively measured by the Morse index) is crucial to answer uniqueness questions. Develop a natural method to compute the Morse index (2016, CAG). (with Graham Smith, Ari Stern, and Detang Zhou) Study the growth of Morse indices of higher dimensional catenoids (2017, Preprint). Characterize the critical catenoid by a natural condition on its Gauss map (2017, Preprint). Hung Tran (TTU) Variational Geometry Feb 20th, / 15
55 Minimal Surfaces Thank You Hung Tran (TTU) Variational Geometry Feb 20th, / 15
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