For a convex domain $D$ that is enclosed by the hypersurface $\partial D$ of bounded normal curvature, we prove an angle comparison theorem for angles between $\partial D$ and geodesic rays starting from some fixed point in $D$, and the corresponding angles for hypersurfaces of constant normal curvature. Also, we obtain a comparison theorem for support functions of such surfaces. As a corollary, we present a proof of Blaschke's Rolling Theorem.
In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some estimates on the higher order mean curvatures of spacelike hypersurfaces satisfying a Omori-Yau maximum principle for certain elliptic operators.
In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is injective and when an immersed horospherically convex hypersurface can be unfolded along the normal flow into an embedded one. These results allow us to establish general Alexandrov reflection principles of elliptic problems of both Weingarten hypersurfaces and complete conformal metrics and relations between them. Consequently, we are able to obtain, for insta…
We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and in a non-positively curved Einstein space. Our results provide additional cases (Theorem 2.3 and Proposition 2.8) that supports the conjecture that a biharmonic submanifold in a sphere has constant mean curvature, and two more cases that support Chen's conjecture on biharmonic h…
We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces (Theorem 3). Thus we generalise [25,Theorem 3] by Nunes. This splitting result follows from an area comparison theorem for hypersurfaces with non-positive Sigma-constant (Theorem 4) that generalises [23, Theorem 2]. Finally, we will address the optimality of these comparison and splitting results by explicitly constructing several examples.
We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof of Hawking's singularity theorem.
Bishop's volume comparison theorem states that a compact $n$-manifold with Ricci curvature larger than the standard $n$-sphere has less volume. While the traditional proof uses geodesic balls, we present another proof using isoperimetric hypersurfaces, also known as "soap bubbles," which minimize area for a given volume. Curiously, isoperimetric hypersurfaces can have codimension 7 singularities, an interesting challenge we are forced to overcome.
In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such suppport hypersurfaces; the results obtained t…
We study properties of non-minimal biharmonic hypersurfaces of spheres. The main result is a CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres. We then deduce new rigidity theorems to support the Conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.
In this paper, we consider an expanding flow of closed, smooth, uniformly convex hypersurface in Euclidean \mathbb{R}^{n+1} with speed u^αf^β(α, β\in\mathbb{R}^1), where u is support function of the hypersurface, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvature radii of the hypersurface. If α\leq 0<β\leq 1-α, we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalization, to a round sphere centered at the origin.
In this paper we discuss an extension of Perelman's comparison for quadrangles. Among applications of this new comparison theorem, we study the equidistance evolution of hypersurfaces in Alexandrov spaces with non-negative curvature. We show that, in certain cases, the equidistance evolution of hypersurfaces become totally convex relative to a bigger sub-domain. An optimal extension of 2nd variational formula for geodesics by Petrunin will be derived for the case of non-negative curvature. In addition, we also introduced the generalized second fundament forms for subsets in Alexandrov spaces. …
For a closed hypersurface $M^n\subset S^{n+1}(1)$ with constant mean curvature and constant non-negative scalar curvature, the present paper shows that if $\mathrm{tr}(\mathcal{A}^k)$ are constants for $k=3,\ldots, n-1$ for shape operator $\mathcal{A}$, then $M$ is isoparametric. The result generalizes the theorem of de Almeida and Brito \cite{dB90} for $n=3$ to any dimension $n$, strongly supporting Chern's conjecture.
Let M be a closed minimal hypersurface in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature. We prove that, if the sum of the cubes of all principal curvatures and the number of distinct principal curvatures are constant, then M is isoparametric. Moreover, We give all possible values for squared length of the second fundamental form of M. This result provides another piece of supporting evidence to the Chern conjecture.
On some specified convex supporting sets of spheres, we find a generalized longitude function whose level sets are totally geodesic. Given an arbitrary (weakly) harmonic map into spheres, the composition of the generalized longitude function and harmonic map satisfies an elliptic equation of divergence type. With the aid of corresponding Harnack inequality, we establish image shrinking property and then the regularity results are followed. Applying such results to study the Gauss image of minimal hypersurfaces in Euclidean spaces, we obtain curvature estimates and corresponding Bernstein theor…
A triharmonic map is a critical point of the tri-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $M^n (n\ge 4)$ is a CMC proper triharmonic hypersurface in a space form $\mathbb{R}^{n+1}(c)$ with four distinct principal curvatures and the multiplicity of the zero principal curvature is at most one, then $M$ has constant scalar curvature. In particular, we obtain any CMC proper triharmonic hypersurface in $\mathbb{R}^5(c)$ is minimal when $c\le 0$, which supports the generalized Chen's conjecture. We also give some characterizations of CMC pr…
In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\mathbb{R}^{n+1}$ with speed $u^αf^{-β}$, where $u$ is the support function of the hypersurface, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. For $α\le 0<β\le 1-α$, we prove that the flow has a unique smooth solution for all time, and converges smoothly after normalization, to a sphere centered at the origin. In particular, the results of Gerhardt \cite{GC3} and Urbas \cite{UJ2} c…
A $k$-harmonic map is a critical point of the $k$-energy in the space of smooth maps between two Riemannian manifolds. In this paper, we prove that if $M^{n} (n\ge 3)$ is a CMC proper triharmonic hypersurface with at most three distinct principal curvatures in a space form $\mathbb{R}^{n+1}(c)$, then $M$ has constant scalar curvature. This supports the generalized Chen's conjecture when $c\le 0$. When $c=1$, we give an optimal upper bound of the mean curvature $H$ for a non-totally umbilical proper CMC $k$-harmonic hypersurface with constant scalar curvature in a sphere. As an application, we …
We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.
We study the evolution of complete non-compact convex hypersurfaces in $\mathbb{R}^{n+1}$ by the inverse mean curvature flow. We establish the long time existence of solutions and provide the characterization of the maximal time of existence in terms of the tangent cone at infinity of the initial hypersurface. Our proof is based on an a'priori pointwise estimate on the mean curvature of the solution from below in terms of the aperture of a supporting cone at infinity. The strict convexity of convex solutions is shown by means of viscosity solutions. Our methods also give an alternative proof o…
We establish volume comparison results for balls in Riemannian manifolds with $C^{1,1}$-metrics with a lower bound on the Ricci tensor and for the evolution of spacelike, acausal, causally complete hypersurfaces with an upper bound on the mean curvature in spacetimes with $C^{1,1}$-metrics with a lower bound on the timelike Ricci curvature. These results are then used to give proofs of Myers' theorem and of Hawking's singularity theorem in this regularity.
We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a comparison principle with geometric barriers for establishing mean curvature estimates for stochastically complete submanifolds in Riemannian products, Riemannian submersions and wedges. These estimates are applied for obtaining both horizontal and vertical half-space theorems for submanifolds in $\mathbb{H}^n \times \mathbb{R}^\ell$.
In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz-Christoffel-Minkowski problem.
For a complete Riemannian manifold $M$ with an (1,1)-elliptic Codazzi self-adjoint tensor field $A$ on it, we use the divergence type operator ${L_A}(u): = div(A\nabla u)$ and an extension of the Ricci tensor to extend some major comparison theorems in Riemannian geometry. In fact we extend theorems like mean curvature comparison theorem, Bishop-Gromov volume comparison theorem, Cheeger-Gromoll splitting theorem and some of their famous topological consequences. Also we get an upper bound for the end of manifolds by restrictions on the extended Ricci tensor. The results can be applicable for s…
We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces, via Reilly's identities. As applications we derive several geometric inequalities for a convex hypersurface $Γ$ in a Cartan-Hadamard manifold $M$. In particular we show that the first mean curvature integral of a convex hypersurface $γ$ nested inside $Γ$ cannot exceed that of $Γ$, which leads to a sharp lower bound in dimension $3$ for the total first mean curvature of $Γ$ in terms of the volume it bounds in $M$. This monotonicity property is extended to all mean curvature integrals when $γ$ is paralle…
In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient manifold are weaker but the assumptions on the surface are considerably more restrictive. We then use our comparison theorem to provide a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature.
This paper continues the study of a class of compact convex hypersurfaces in Euclidean space $R^{n+1}, ~n \geq 1$, which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In $R^3$ reflectors arise naturally in geometrical optics and are used in design of light reflectors and reflector antennas. They are also important in rendering problems in computer graphics. The notion of a focal function for reflectors plays a central role similar to that of the Minkowski support function for …
The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds of arbitrary index, using one-sided bounds on the Riemann tensor which in the Riemannian case correspond to one-sided bounds on the sectional curvatures. Starting from 2-dimensional rigidity results and using an inductive technique, a new class of gap-type rigidity theorems is proved for semi-Riemannian manifolds of arbitrary index, generalizing those first given by Gromov and Greene-Wu. As applications we prove rigidity results for semi-Riemannia…
In this paper, we study a class of fully nonlinear contracting curvature flows of closed, uniformly convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with the normal speed $Φ$ given by $r^αF^β$ or $u^αF^β$, where $F$ is a monotone, symmetric, inverse-concave, homogeneous of degree one function of the principal curvatures, $r$ is the distance from the hypersurface to the origin and $u$ is the support function of hypersurface. If $α\geq β+1$ when $Φ=r^αF^β$ or $α> β+1$ when $Φ=u^αF^β$, we prove that the flow exists for all times and converges to the origin. After proper rescaling, we…
We consider $n$-dimensional hypersurfaces flowing by mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. We show that the Hausdorff $n$-measure of the singular set is zero. In fact, we consider two types of interaction between the support and flowing surfaces. In the case of weaker interaction, we need make no further assumptions than in the case without boundary to achieve our result. In the case of stronger interaction, we need only make the additional assumption that $H_Σ>0$, that is, that the support surface be mean convex. We go on, in this cas…
We consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_k^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_k is the k-th symmetric polynomial of the principle curvature radii of the hypersurface, k is an integer and 1<= k<= n. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of a elliptic equation, when the constants alpha, beta bel…
We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature radii of the hypersurface. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of an elliptic equation, when the constants alpha, beta belong to a suitable range, provid…
We single out a notion of staticity which applies to any domain in hyperbolic space whose boundary is a non-compact totally umbilical hypersurface. For (time-symmetric) initial data sets modeled at infinity on any of these latter examples, we formulate and prove a positive mass theorem in the spin category under natural dominant energy conditions (both in the interior and along the boundary) whose rigidity statement retrieves, among other things, a sharper version of a recent result by Souam to the effect that no such hypersurface admits a compactly supported deformation keeping the original l…
