東華大學圖書館 |

Min-Max Construction of Minimal Submanifolds and Scalar Curvature Comparison and Rigidity /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Min-Max Construction of Minimal Submanifolds and Scalar Curvature Comparison and Rigidity // Dongyeong Ko.
作者:	Ko, Dongyeong,
面頁冊數:	1 electronic resource (263 pages)
附註:	Source: Dissertations Abstracts International, Volume: 87-01, Section: B.
Contained By:	Dissertations Abstracts International87-01B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31932744
ISBN:	9798288805226

Min-Max Construction of Minimal Submanifolds and Scalar Curvature Comparison and Rigidity /
Ko, Dongyeong,

Min-Max Construction of Minimal Submanifolds and Scalar Curvature Comparison and Rigidity /Dongyeong Ko. - 1 electronic resource (263 pages)

Source: Dissertations Abstracts International, Volume: 87-01, Section: B.

Minimal submanifolds are fundamental objects in differential geometry, which represents submanifolds which locally minimizes area. Not limited to the existence theory of minimal submanifolds possesses the important itself, the theory has led the progress to many fundamental results in differential geometry by enormous applications. In this thesis, we discuss existence and Morse-theoretic characterization of minimal sub-manifolds in various setting, and prove a scalar curvature comparison and rigidity results as an application. The first part will be mainly devoted to the existence theory of min-max embedded geodesics and Morse theoretic characterization. In Chapter 1, we give a Morse-theoretic characterization of simple closed geodesics on Riemannian 2-spheres. On any Riemannian 2-sphere endowed with a generic metric, we show there exists a simple closed geodesic with Morse index 1, 2 and 3. In particular, for an orientable Riemannian surface we prove strong Morse inequalities for the length functional applied to the space of simple closed curves.In Chapter 2, we prove the existence of two capillary embedded geodesics with a contact angle θ ∈ (0, π/2) on Riemannian 2-disks with strictly convex boundary, where the absence of a simple closed geodesic loop based on a point of boundary is given. In particular, our condition contains the cases of Riemannian 2-disks with strictly convex boundary, nonnegative Gaussian curvature and total geodesic curvature lower bound π of the boundary. Moreover, by providing examples, we prove that our total geodesic curvature condition is sharp to admit a capillary embedded geodesic with a contact angle θ ∈ (0, π/2) under the nonnegative interior Gaussian curvature condition. We also prove the existence of Morse Index 1 and 2 capillary embedded geodesics by applying ideas in Chapter 1.In Chapter 3, we prove that a free boundary curve shortening flow on closed surfaces with a strictly convex boundary remains non-collapsed for a finite time in the sense of the reflected chord-arc profile introduced by Langford-Zhu. This shows that such flow converges to free boundary embedded geodesic in infinite time, or shrinks to a round half-point on the boundary. As a consequence, we prove the existence of two free boundary embedded geodesics on a Riemannian 2-disk with a strictly convex boundary. Moreover, we prove that there exists a free boundary embedded geodesic with Morse Index 1 and 2. This settles the free boundary analog of Grayson's theorem.In the second part, in Chapter 5, we will focus on the comparison and rigidity of manifolds with positive scalar curvature with boundary based on minimal surface theory, which is based on the joint work with Xuan Yao. In Chapter 4, for a compact Riemannian 3-manifold (M3, g) with mean convex boundary which is diffeomorphic to a weakly convex compact domain in R3 , we prove that if scalar curvature is nonnegative and the scaled mean curvature comparison H2g ≥ H20gEucl holds then (M, g) is flat. Our result is a smooth analog of Gromov's dihedral rigidity conjecture and an effective version of extremality results on weakly convex balls in R3. More generally, we prove the comparison and rigidity theorem for several classes of manifold with corners. Our proof uses capillary minimal surfaces with prescribed contact angle together with the construction of foliation with nonnegative mean curvature and with prescribed contact angles.In the last part, in Chapter 6, we develop equivariant min-max theory to construct minimal hypersurfaces on manifolds with symmetry. We develop cohomogeneity two equivariant min-max theory for minimal hypersurfaces proposed by Pitts and Rubinstein in 1980s. We expect the theory may produce minimal hypersurfaces with arbitrary large Betti numbers on manifolds with symmetry.

English

ISBN: 9798288805226Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Min-max theory

Min-Max Construction of Minimal Submanifolds and Scalar Curvature Comparison and Rigidity /
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520 $a Minimal submanifolds are fundamental objects in differential geometry, which represents submanifolds which locally minimizes area. Not limited to the existence theory of minimal submanifolds possesses the important itself, the theory has led the progress to many fundamental results in differential geometry by enormous applications. In this thesis, we discuss existence and Morse-theoretic characterization of minimal sub-manifolds in various setting, and prove a scalar curvature comparison and rigidity results as an application. The first part will be mainly devoted to the existence theory of min-max embedded geodesics and Morse theoretic characterization. In Chapter 1, we give a Morse-theoretic characterization of simple closed geodesics on Riemannian 2-spheres. On any Riemannian 2-sphere endowed with a generic metric, we show there exists a simple closed geodesic with Morse index 1, 2 and 3. In particular, for an orientable Riemannian surface we prove strong Morse inequalities for the length functional applied to the space of simple closed curves.In Chapter 2, we prove the existence of two capillary embedded geodesics with a contact angle θ ∈ (0, π/2) on Riemannian 2-disks with strictly convex boundary, where the absence of a simple closed geodesic loop based on a point of boundary is given. In particular, our condition contains the cases of Riemannian 2-disks with strictly convex boundary, nonnegative Gaussian curvature and total geodesic curvature lower bound π of the boundary. Moreover, by providing examples, we prove that our total geodesic curvature condition is sharp to admit a capillary embedded geodesic with a contact angle θ ∈ (0, π/2) under the nonnegative interior Gaussian curvature condition. We also prove the existence of Morse Index 1 and 2 capillary embedded geodesics by applying ideas in Chapter 1.In Chapter 3, we prove that a free boundary curve shortening flow on closed surfaces with a strictly convex boundary remains non-collapsed for a finite time in the sense of the reflected chord-arc profile introduced by Langford-Zhu. This shows that such flow converges to free boundary embedded geodesic in infinite time, or shrinks to a round half-point on the boundary. As a consequence, we prove the existence of two free boundary embedded geodesics on a Riemannian 2-disk with a strictly convex boundary. Moreover, we prove that there exists a free boundary embedded geodesic with Morse Index 1 and 2. This settles the free boundary analog of Grayson's theorem.In the second part, in Chapter 5, we will focus on the comparison and rigidity of manifolds with positive scalar curvature with boundary based on minimal surface theory, which is based on the joint work with Xuan Yao. In Chapter 4, for a compact Riemannian 3-manifold (M3, g) with mean convex boundary which is diffeomorphic to a weakly convex compact domain in R3 , we prove that if scalar curvature is nonnegative and the scaled mean curvature comparison H2g ≥ H20gEucl holds then (M, g) is flat. Our result is a smooth analog of Gromov's dihedral rigidity conjecture and an effective version of extremality results on weakly convex balls in R3. More generally, we prove the comparison and rigidity theorem for several classes of manifold with corners. Our proof uses capillary minimal surfaces with prescribed contact angle together with the construction of foliation with nonnegative mean curvature and with prescribed contact angles.In the last part, in Chapter 6, we develop equivariant min-max theory to construct minimal hypersurfaces on manifolds with symmetry. We develop cohomogeneity two equivariant min-max theory for minimal hypersurfaces proposed by Pitts and Rubinstein in 1980s. We expect the theory may produce minimal hypersurfaces with arbitrary large Betti numbers on manifolds with symmetry.
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