Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space (Q6567151)

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 corresponding solution \(Y(M^{n},t)\subset B_{1}(0)\subset \mathbb{R}^{n+1}\) and its support function \(s(z,t)\) satisfy the equations \N\[\N\frac{\partial }{ \partial t}Y=\sqrt{(1-\left\vert Y\right\vert ^{2})(1-\left\langle N,Y\right\rangle ^{2})}(\phi (t)-(K^{X})^{\alpha })N,\N\] \N\[\N\frac{\partial }{ \partial t}s=\sqrt{(1-s^{2}-\left\vert \overline{\nabla }s\right\vert ^{2})(1-s^{2})}(\phi (t)-(K^{X})^{\alpha }),\N\] \Nwhere the Gauss curvature \( K^{X}\) is related to the Gauss curvature \(K^{Y}=\frac{1}{\det (\tau _{ij})}\) , with \(\tau _{ij}=\overline{\nabla }_{i}\overline{\nabla }_{j}s-s\sigma _{ij}\), \(\overline{\nabla }\) is the gradient with respect to the round metric \(\sigma \) on \(\mathbb{S}^{n}\), and \(N\) is the Gauss map. The authors prove that convexity is preserved along the flow, they prove a priori \(C^{0}\) and \(C^{1}\) estimates for the flow and the projected flow, a positive lower bound for the principal curvatures along the flow, and an upper bound for the Gauss curvature on any finite time interval. They prove by contradiction the long time existence of the flow and the subsequential Hausdorff convergence of \(M_{t}\) and the convergence of the center of the inner ball of \(\Omega _{t}\) to a fixed point.