Mean Curvature Flow with Bounded Gauss Image

We study the mean curvature flow of a complete space-like submanifold in pseudo-Euclidean space with bounded Gauss image and bounded curvature. We establish a relevant maximum principle for our setting. Then, we can obtain the ??confinable property?? of the Gauss images and curvature estimates under the mean curvature flow. Thus we prove a corresponding long time existence result.     

Curvature Estimates of Hypersurfaces in the Minkowski Space

A class of curvature estimates of spacelike admissible hypersurfaces related to translating solitons of the higher order mean curvature flow in the Minkowski space is obtained,which may ofer an idea to study an open question of the existence of hypersurfaces with the prescribed higher mean curvature in the Minkowski space.     

Volume estimate for a cone with a submanifold as vertex

We give some estimates for the volume of a cone with vertex a submanifold P of a Riemannian or Kaehler manifold M. The estimates are functions of bounds of the mean curvature of P and the sectional curvature of M. They are sharp on cones having a basis which is contained in a tubular hypersurface about P in a space form or in a complex space form.Work partially supported by DGICYT Grant No. PB90-0014-C03-01.     

Mean Curvature Flow via Convex Functions on Grassmannian Manifolds

Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow have been obtained, which reveal that if the Gauss image of the initial submanifold is contained in a certain sublevel set of the v-function, then all the Gauss images of the submanifolds under the mean curvature flow are also contained in the same sublevel set of the v-function. Under such restrictions, curvature estimates in terms of v-function composed with the Gauss map can be carried out.     

General curvature estimates for stable H-surfaces immersed into a space form

In this paper, we give general curvature estimates for constant mean curvature surfaces immersed into a simply-connected 3-dimensional space form. We obtain bounds on the norm of the traceless second fundamental form and on the Gaussian curvature at the center of a relatively compact stable geodesic ball (and, more generally, of a relatively compact geodesic ball with stability operator bounded from below). As a by-product, we show that the notions of weak and strong Morse indices coincide for complete non-compact constant mean curvature surfaces. We also derive a geometric proof of the fact that a complete stable surface with constant mean curvature 1 in the usual hyperbolic space must be a horosphere.     

Height estimates and half-space type theorems in weighted product spaces with nonnegative Bakry–Émery–Ricci curvature

We prove height estimates concerning compact hypersurfaces with nonzero constant weighted mean curvature and whose boundary is contained into a slice of a weighted product space of nonnegative Bakry–Émery–Ricci curvature. As applications of our estimates, we obtain half-space type results related to complete noncompact hypersurfaces properly immersed in such an ambient space.     

Ricci Curvature and Bochner Formulas for Martingales

We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two‐sided bounds on Ricci curvature in much the same manner that the classical Bochner formula on M is related to lower bounds on Ricci curvature. Using this formalism, we obtain new characterizations of bounded Ricci curvature, new gradient estimates for martingales on path space, new Hessian estimates for martingales on path space, and streamlined proofs of the previous characterizations of bounded Ricci curvature.© 2018 Wiley Periodicals, Inc.     

On an extension of the H k mean curvature flow

In this note,we generalize an extension theorem in [Le-Sesum] and [Xu-Ye-Zhao] of the mean curvature flow to the Hk mean curvature flow under some extra conditions.The main difficulty in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the Hk mean curvature flow,and to do a suitable Moser iteration process.These two problems are overcome by imposing some extra conditions which may be weakened or removed in our forthcoming paper.On the other hand,we derive some estimates for the generalized mean curvature flow,which have their own interesting.     

Non-preserved curvature conditions under constrained mean curvature flows

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.     

Mean Curvature Flow with Convex Gauss Image

In this paper,the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is studied.The confinable property of the Gauss image under the mean curvature flow is proved,which in turn helps one to obtain the curvature estimates.Then the author proves a long time existence result.The asymptotic behavior of these solutions when t→∞is also studied.     

On the Curvatures of Complete Spacelike Hypersurfaces in de Sitter Space

We establish some estimates for the higher-order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in de Sitter space which is contained in certain unbounded regions of the ambient space. Our results will be an application of a generalized maximum principle due to Omori.     

Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries

We consider the area-preserving mean curvature flow with free Neumann boundaries. We show that a rotationally symmetric n-dimensional hypersurface in R~(n+1)between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the maximal principle to obtain gradient and curvature estimates, leading to long-time existence of the flow and convergence to a constant mean curvature surface.     

Curvature Estimates for Strongly Stable Domains on Surfaces with Constant Mean Curvature in Space Forms

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Some a priori bounds for solutions of the constant Gauss curvature equation

In this work, we give a priori height and gradient estimates for solutions of the prescribed constant Gauss curvature equation in Euclidean space. We shall consider convex radial graphs with positive constant mean curvature. The estimates are established by considering in such a graph, the Riemannian metric given by the second fundamental form of the immersion.     

Hyperbolic mean curvature flow

In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations is strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth solution and possesses the nonlinear stability defined on the Euclidean space with dimension larger than 4. We derive nonlinear wave equations satisfied by some geometric quantities related to the hyperbolic mean curvature flow. Moreover, we also discuss the relation between the equations for hyperbolic mean curvature flow and the equations for extremal surfaces in the Minkowski space-time.     

Convexity estimates for a nonhomogeneous mean curvature flow

We study the evolution of a closed immersed hypersurface whose speed is given by a function f(H){\phi(H)} (H) of the mean curvature asymptotic to H/ ln H for large H. Compared with other nonlinear functions of the curvatures, this speed has some good properties which allow for an easier study of the formation of singularities in the nonconvex case. We prove apriori estimates showing that any surface with positive mean curvature at the initial time becomes asymptotically convex near a singularity. Similar estimates also hold for the mean curvature flow; for the flow considered here they admit a simpler proof based only on the maximum principle.     

Spectral gap on Riemannian path space over static and evolving manifolds

In this article, we continue the discussion of Fang–Wu (2015) to estimate the spectral gap of the Ornstein–Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along with explicit estimates we study the short-time asymptotics of the spectral gap. The results are then extended to the path space of Riemannian manifolds evolving under a geometric flow. Our paper is strongly motivated by Naber's recent work (2015) on characterizing bounded Ricci curvature through stochastic analysis on path space.     

A Relation between Mean Curvature Flow Solitons and Minimal Submanifolds

We derive a one to one correspondence between conformal solitons of the mean curvature flow in an ambient space N and minimal submanifolds in a different ambient space where equals ℝ × N equipped with a warped product metric and show that a submanifold inN converges to a conformal soliton under the mean curvature flow in N if and only if its associatedsubmanifold in converges to a minimal submanifold under a rescaled mean curvature flow in . We then define a notion of stability for conformal solitons and obtain L^p estimates as well as pointwise estimates for the curvature of stable solitons.     

The blow-up of the conformal mean curvature flow

In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space R~n. This kind of flow is a special case of a general modified mean curvature flow which is of various origination. As the main result, we prove a blow-up theorem concluding that, under the conformal mean curvature flow in R~n, the maximum of the square norm of the second fundamental form of any compact submanifold tends to infinity in finite time. Furthermore, we also prove that the external conformal forced mean curvature flow of a compact submanifold in R~n with the same pinched condition as Andrews-Baker's will be convergent to a round point in finite time.