Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space
type: Item
instance of: scholarly article
title: Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space
zbMATH Open document ID: 1545.53091
author: Yong Wei
author: Bo Yang
author: Tailong Zhou
published in: Transactions of the American Mathematical Society
publication date: 2024-07-04T00:00:00Z
review text: The authors consider a smooth embedding \(X_{0}:M^{n}\rightarrow \mathbb{H} ^{n+1}\), \(n\geq 2\), such that \(M_{0}=X_{0}(M)\) is a closed and convex hypersurface in the hyperbolic space \N\[\N\mathbb{H}^{n+1}=\{X\in \mathbb{R} ^{1,n+1}\mid \left\langle X,X\right\rangle =-1, X_{0}>0\},\N\] \Nand a smooth family of embeddings \(X:M^{n}\times \lbrack 0,T)\rightarrow \mathbb{H}^{n+1}\) satisfying \N\[\N\frac{\partial }{\partial t}X(x,t)=(\phi (t)-K^{\alpha })\nu (x,t), X(\cdot ,0)=X_{0}(\cdot ),\N\] \Nwhere \(\alpha >0\), \(\nu \) is the unit outer normal of \(M_{t}=X(M,t)\), \(K\) is the Gauss curvature of \(M_{t}\) and \( \phi \) is defined through \N\[\N\phi (t)=\frac{1}{\left\vert M_{t}\right\vert } \int_{M_{t}}K^{\alpha }d\mu _{t},\N\] \Nsuch that the domain \(\Omega _{t}\) enclosed by \(M_{t}\) has a fixed volume \(\left\vert \Omega _{t}\right\vert =\left\vert \Omega _{0}\right\vert \) along the flow. \N\NThe main result of the paper proves that for any \(\alpha >0\), the volume preserving flow has a unique smooth convex solution \(M_{t}\) for all time \(t\in \lbrack 0,\infty )\) , and the solution \(M_{t}\) converges smoothly and exponentially as \( t\rightarrow \infty \) to a geodesic sphere of radius \(\rho _{\infty }\) which encloses the same volume as \(M_{0}\). For the proof, the authors first recall properties of the geometry of hypersurfaces in hyperbolic space, the evolution equations along the flow, and the quermassintegrals and curvature measures in the hyperbolic space. They project the flow to the Euclidean space via the Klein model, which parametrizes the hyperbolic space \(\mathbb{H }^{n+1}\) using the unit disc, through a projection from an embedding \( X:M^{n}\rightarrow \mathbb{H}^{n+1}\) to an embedding \(Y:M^{n}\rightarrow B_{1}(0)\) by \(X=\frac{(1,Y)}{\sqrt{1-\left\vert Y\right\vert ^{2}}}\). If \( X(M^{n},t)\subset \mathbb{H}^{n+1}\), \(t\in \lbrack 0,T)\) is a smooth convex solution to the flow, then up to a tangential diffeomorphism, the…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 53E20
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 7876034
zbMATH Keywords: volume-preserving Gauss curvature flow
zbMATH Keywords: hyperbolic space
zbMATH Keywords: curvature measures
zbMATH Keywords: Alexandrov reflection
zbMATH Keywords: convexity preservation
zbMATH Keywords: long time existence
MaRDI profile type: Publication
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DOI: 10.1090/TRAN/9095
description: scientific article; zbMATH DE number 7876034