Contraction of surfaces in hyperbolic space and in sphere
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Calculus of Variations
Contraction of surfaces in hyperbolic space and in sphere Yingxiang Hu1 · Haizhong Li2 · Yong Wei3 · Tailong Zhou3 Received: 27 November 2019 / Accepted: 31 July 2020 / Published online: 16 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we consider the contracting curvature flows of smooth closed surfaces in 3dimensional hyperbolic space and in 3-dimensional sphere. In the hyperbolic case, we show that if the initial surface M0 has positive scalar curvature, then along the flow by a positive power α of the mean curvature H , the evolving surface Mt has positive scalar curvature for t > 0. By assuming α ∈ [1, 4], we can further prove that Mt contracts a point in finite time and become spherical as the final time is approached. We also show the same conclusion for the flows by powers of scalar curvature and by powers of Gauss curvature provided that the power α ∈ [1/2, 1]. In the sphere case, we show that the flow by a positive power α of mean curvature contracts strictly convex surface in S3 to a round point in finite time if α ∈ [1, 5]. The same conclusion also holds for the flow by powers of Gauss curvature provided that the power α ∈ [1/2, 1]. Mathematics Subject Classification 53C44 · 53C21
Communicated by J. Jost.
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Yingxiang Hu [email protected] Haizhong Li [email protected] Yong Wei [email protected] Tailong Zhou [email protected]
1
School of Mathematical Sciences, Beihang University, Beijing 100191, People’s Republic of China
2
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
3
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, People’s Republic of China
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1 Introduction Let R3 (c) (c = 0, 1, −1) be the standard model of simply connected space form, i.e., when c = 0, R3 (0) = R3 , when c = 1, R3 (1) = S3 and when c = −1, R3 (−1) = H3 . Let M0 be a smooth closed surface in R3 (c), given by a smooth immersion X 0 : M 2 → R3 (c). We consider the contracting curvature flow of closed surfaces starting at M0 in R3 (c), which is a family of smooth immersions X : M 2 × [0, T ) → R3 (c) satisfying ∂ ∂t X (x, t) = −F(x, t)ν(x, t), (1.1) X (x, 0) = X 0 (x), where F is a smooth, symmetric function of the principal curvatures κ = (κ1 , κ2 ) of the evolving surface Mt = X (M, t), and ν is the outward unit normal of Mt .
1.1 Background There are many papers which consider the contraction of hypersurfaces in Euclidean space Rn+1 under the flow (1.1). In his foundational work [22], Huisken proved that any compact strictly convex hypersurface in Euclidean space, evolving by the mean curvature flow (i.e., F is given by the mean curvature H ), will become spherical as it shrinks to a point. Later, Chow [16] proved the same conclusion for flow (1.1) with speed F given by nth root of the Gauss curvature K . He also proved a result for flow by the square root of the scalar curvature [17], but in t
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