Volume preserving curvature flows in Lorentzian manifolds

Let N be a (n + 1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface ${\mathcal{S}_{0}}$ and F a curvature function, either the mean curvature H, the root of the second symmetric polynomial ${{\sigma}_{2}=\sqrt{H_{2}}}$ or a curvature function of class (K*), a class of curvature functions which includes the nth root of the Gaussian curvature ${{\sigma}_{n}= K^{\frac{1}{n}}}$ . We consider curvature flows with curvature function F and a volume preserving term and prove long time existence of the flow and exponential convergence of the corresponding graphs in the C ^∞-topology to a hypersurface of constant F-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfaces.     

Local well‐posedness for volume‐preserving mean curvature and Willmore flows with line tension

We show the short‐time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by the volume‐preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non‐local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a curvilinear cordinate system due to Vogel to write the flows as graphs over a fixed reference hypersurface.     

A note on inverse mean curvature flow in cosmological spacetimes

In 12 Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Furthermore, it is shown in 12 that the leaves of the inverse mean curvature flow provide a foliation of the future of the initial hypersurface.We show that this result persists, if we generalize the setting by leaving the mean curvature barrier assumption out. For initial hypersurfaces with sufficiently large mean curvature we can weaken the timelike convergence condition to a physically relevant energy condition.     

Stability and geometric properties of constant weighted mean curvature hypersurfaces in gradient Ricci solitons

In this paper, we study stability properties of hypersurfaces with constant weighted mean curvature (CWMC) in gradient Ricci solitons. The CWMC hypersurfaces generalize the f-minimal hypersurfaces and appear naturally in the isoperimetric problems in smooth metric measure spaces. We obtain a result about the relationship between the properness and extrinsic volume growth under the assumption of a limitation for the weighted mean curvature of the immersion. Moreover, we estimate Morse index for CWMC hypersurfaces in terms of the dimension of the space of parallel vector fields restricted to hypersurface.     

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.     

Sufficient conditions for constant mean curvature surfaces to be round

We study under what condition a constant mean curvature surface can be round: i) If the boundary of a compact immersed disk type constant mean curvature surface in consists of lines of curvature and has less than 4 vertices with angle , then the surface is spherical; ii) A compact immersed disk type capillary surface with less than 4 vertices in a domain of bounded by spheres or planes is spherical; iii) The mean curvature vector of a compact embedded capillary hypersurface of with smooth boundary in an unbounded polyhedral domain with unbalanced boundary should point inward; iv) If the kth order () mean curvature of a compact immersed constant mean curvature hypersurface of without boundary is constant, then the hypersurface is a sphere. Received: 3 October 2000 / Published online: 1 February 2002     

Une nouvelle caractérisation des sphères géodésiques dans les espaces modèles

We show that a compact immersed hypersurface of hyperbolic space or an open half-sphere with constant mean curvature and almost constant scalar curvature is a geodesic sphere. To cite this article: J. Roth, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Convex mean curvature flow with a forcing term in direction of the position vector

A smooth, compact and strictly convex hypersurface evolving in ℝ ⁿ⁺¹ along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough. The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper. Long-time existence and convergence of normalization of the flow are also investigated.     

A note on compact Weingarten hypersurfaces embedded in $$\mathbb {R}^{n + 1}$$

We prove that the round sphere is the only compact Weingarten hypersurface embedded in the Euclidean space such that \(H_r = aH + b\), for constants \(a, b \in \mathbb {R}\). Here, \(H_r\) stands for the r-th mean curvature and H denotes the standard mean curvature of the hypersurface.     

On extrinsic symmetric spaces with zero mean curvature in Minkowski space-time

For an extrinsic symmetric space M in Minkowski space-time, we prove that if M is spacelike with zero mean curvature, then it is totally geodesic and if M is timelike with zero mean curvature, then it is totally geodesic or it is a flat hypersurface.     

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].     

Caccioppoli’s inequalities on constant mean curvature hypersurfaces in Riemannian manifolds

We prove some Caccioppoli’s inequalities for the traceless part of the second fundamental form of a complete, noncompact, finite index, constant mean curvature hypersurface of a Riemannian manifold, satisfying some curvature conditions. This allows us to unify and clarify many results scattered in the literature and to obtain some new results. For example, we prove that there is no stable, complete, noncompact hypersurface in ${{\mathbb R}^{n+1}, n \leq 5}$ , with constant mean curvature ${H \not=0}$ , provided that, for suitable p, the L ^p norm of the traceless part of second fundamental form satisfies some growth condition.     

The Willmore functional and the containment problem in R4

Let Σ be a convex hypersurface in the Euclidean space R ⁴ with mean curvature H. We obtain a geometric lower bound for the Willmore functional ∫_Σ H ² dσ. This bound is an invariant involving the area of Σ, the volume and Minkowski quermassintegrals of the convex body that Σ bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R ⁴.     

On the nonexistence of a hypersurface of prescribed mean curvature with a given boundary

A general method is developed for finding necessary conditions for a given codimension-two submanifold Γ of a riemannian manifold to be the boundary of an immersed hypersurface of prescribed mean curvature. In the simplest case the condition is a comparison of the magnitude of the mean curvature with the ratio of the volume of the projection of Γ into a hyperplane to the volume of the interior of that projection. The method is applied to show that certain recent existence results for surfaces of prescribed mean curvature may not be quantitatively improved.     

Contact Hypersurfaces of a Bochner–Kaehler Manifold

We have studied contact metric hypersurfaces of a Bochner–Kaehler manifold and obtained the following two results: (1) a contact metric constant mean curvature (CMC) hypersurface of a Bochner–Kaehler manifold is a (k, μ)-contact manifold, and (2) if M is a compact contact metric CMC hypersurface of a Bochner–Kaehler manifold with a conformal vector field V that is neither tangential nor normal anywhere, then it is totally umbilical and Sasakian, and under certain conditions on V, is isometric to a unit sphere.     

Total mean curvature of positively curved hypersurfaces

In this article, we prove that every positively curved, complete non-compact hypersurface in Rn has infinite total mean curvature.     

On the structure of complete hypersurfaces in a Riemannian manifold of nonnegative curvature and L 2 harmonic forms

Let M ⁿ be a complete oriented noncompact hypersurface in a complete Riemannian manifold N ⁿ⁺¹ of nonnegative sectional curvature with ${2 \leq n \leq 5}$ . We prove that if M satisfies a stability condition, then there are no non-trivial L ² harmonic one-forms on M. This result is a generalization of a well-known fact in the case when M is a stable minimally immersed hypersurface. As a consequence, we show that if the mean curvature of M is constant, then either M must have only one end or M splits into a product of ${\mathbb{R}}$ and a compact manifold with nonnegative sectional curvature. In case ${n \geq 5}$ , we also show that the same result holds if the absolute value of the mean curvature is less than or equal to the ratio of the norm of the second fundamental form to the dimension of a hypersurface.     

On the problem of isometry of a hypersurface preserving mean curvature

The problem of determining the Bonnet hypersurfaces in R ⁿ⁺¹, for n > 1, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/or (the locus of) itself is called Bonnet associate of the initial hypersurface. The orthogonal net which is called A-net is special and very important for our study and it is described on a hypersurface. It is proved that, non-minimal hypersurface in R ⁿ⁺¹ with no umbilical points is a Bonnet hypersurface if and only if it has an A-net.     

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.     

Complete hypersurfaces immersed in a semi-Riemannian warped product

The aim of this paper is to study the uniqueness of complete hypersurfaces immersed in a semi-Riemannian warped product whose warping function has convex logarithm and such that its fiber has constant sectional curvature. By using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds and supposing a natural comparison inequality between the r-th mean curvatures of the hypersurface and that ones of the slices of the region where the hypersurface is contained, we are able to prove that a such hypersurface must be, in fact, a slice.