Alexandrov's Theorem Research Articles

On effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes

We study the effects of the null energy condition on totally umbilic hypersurfaces in a class of static spacetimes, both in the spacelike and the timelike case, respectively. In the spacelike case, we study totally umbilic warped product graphs and give a full characterization of embedded surfaces with constant spacetime mean curvature using an Alexandrov Theorem by Brendle and Borghini–Fogagnolo–Pinamonti. In the timelike case, we achieve a characterization of photon surfaces with constant umbilicity factor similar to a result by Cederbaum–Galloway.

A geometric approach to second-order differentiability of convex functions

We show a new, elementary and geometric proof of the classical Alexandrov theorem about the second order differentiability of convex functions. We also show new proofs of recent results about Lusin approximation of convex functions and convex bodies by C 1 , 1 C^{1,1} convex functions and convex bodies.

Moving plane method for varifolds and applications

abstract: In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compactly supported varifolds with smooth boundary and in the setting of varifolds without boundary. A key ingredient is a Hopf lemma for stationary varifolds and varifolds of constant mean curvature. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compactly supported varifolds of constant mean curvature in hyperbolic space.

Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow

Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow

Cut locus realizations on convex polyhedra

We prove that every positively weighted tree T can be realized as the cut locus C(x) of a point x on a convex polyhedron P, with T edge weights matching C(x) edge lengths. If T has n leaves, P has (in general) n+1 vertices. We show there is in fact a continuum of polyhedra P each realizing T for some x∈P. Three main tools in the proof are properties of the star unfolding of P, Alexandrov's gluing theorem, and a new cut-locus partition lemma. The construction of P from T is surprisingly simple.

Asymptotic of the discrete volume-preserving fractional mean curvature flow via a nonlocal quantitative Alexandrov theorem

We characterize the long time behaviour of a discrete-in-time approximation of the volume preserving fractional mean curvature flow. In particular, we prove that the discrete flow starting from any bounded set of finite fractional perimeter converges exponentially fast to a single ball if the dimension N≤7 and the fractional esponent s≈1, or for any s∈(0,1) when N=2. As an intermediate result we establish a fractional quantitative Alexandrov type estimate for normal deformations of a ball. Finally, we provide existence for flat flows as limit points of the discrete flow when the time discretization parameter tends to zero. Furthermore, analogous results for the classical perimeter are obtained in the limit s→1.

Alexandrov theorem for general nonlocal curvatures: The geometric impact of the kernel

For a general radially symmetric, non-increasing, non-negative kernel h∈Lloc1(Rd), we study the rigidity of measurable sets in Rd with constant nonlocal h-mean curvature. Under a suitable “improved integrability” assumption on h, we prove that these sets are finite unions of equal balls, as soon as they satisfy a natural nondegeneracy condition. Both the radius of the balls and their mutual distance can be controlled from below in terms of suitable parameters depending explicitly on the measure of the level sets of h. In the simplest, common case, in which h is positive, bounded and decreasing, our result implies that any bounded open set or any bounded measurable set with finite perimeter which has constant nonlocal h-mean curvature has to be a ball.

New integral estimates in substatic Riemannian manifolds and the Alexandrov Theorem

We derive new integral estimates on substatic manifolds with boundary of horizon type, naturally arising in General Relativity. In particular, we generalize to this setting an identity due to Magnanini-Poggesi [24] leading to the Alexandrov Theorem in Rn and improve on a Heintze-Karcher type inequality due to Li-Xia [22]. Our method relies on the introduction of a new vector field with nonnegative divergence, generalizing to this setting the P-function technique of Weinberger [36].

Characterizations of umbilic hypersurfaces in warped product manifolds

We consider the closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using an integral formula or Brendle’s Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.

Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms

<p style='text-indent:20px;'>We study a level-set mean curvature flow equation with driving and source terms, and establish convergence results on the asymptotic behavior of solutions as time goes to infinity under some additional assumptions. We also study the associated stationary problem in details in a particular case, and establish Alexandrov's theorem in two dimensions in the viscosity sense, which is of independent interest.

An Alexandrov theorem in Minkowski spacetime

In this paper, we generalize a theorem {a} la Alexandrov of Wang, Wang and Zhang [WWZ] for closed codimension-two spacelike submanifolds in the Minkowski spacetime for an adapted CMC condition .

Secondary Polytope and Secondary Power Diagram

An ingenious construction of Gel’fand, Kapranov, and Zelevinsky [5] geometrizes the triangulations of a point configuration, such that all coherent triangulations form a convex polytope, the so-called secondary polytope. The secondary polytope can be treated as a weighted Delaunay triangulation in the space of all possible coherent triangulations. Naturally, it should have a dual diagram. In this work, we explicitly construct the secondary power diagram, which is the power diagram of the space of all possible power diagrams with nonempty boundary cells. Secondary power diagram gives an alternative proof for the classical secondary polytope theorem based on Alexandrov theorem. Furthermore, secondary power diagram theory shows one can transform a nondegenerated coherent triangulation to another nondegenerated coherent triangulation by a sequence of bistellar modifications, such that all the intermediate triangulations are nondegenerated and coherent. As an application of this theory, we propose an algorithm to triangulate a special class of 3d nonconvex polyhedra without using additional vertices. We prove that this algorithm terminates in $$O({{n}^{3}})$$ time.

Bumpy pyramid folding

We investigate folding problems for a class of petal (or star-like) polygons P, which have an n-polygonal base B surrounded by a sequence of triangles for which adjacent pairs of sides have equal length. We give linear time algorithms using constant precision to determine if P can fold to a pyramid with flat base B, and to determine a triangulation of B (crease pattern) that allows folding into a convex (triangulated) polyhedron. By Alexandrov's Theorem, the crease pattern is unique if it exists, but the general algorithm known for this theorem is pseudo-polynomial, with running time O(n456.5); ours is the first efficient algorithm for Alexandrov's Theorem for a special class of polyhedra. We also give a polynomial time algorithm that finds the crease pattern to produce the maximum volume triangulated polyhedron.

A sharp quantitative version of Alexandrov's theorem via the method of moving planes

We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let S be a C^2 closed embedded hypersurface of \mathbb{R}^{n+1} , n\geq1 , and denote by osc (H) the oscillation of its mean curvature. We prove that there exists a positive \varepsilon , depending on n and upper bounds on the area and the C^2 -regularity of S , such that if osc (H) \leq \varepsilon then there exist two concentric balls B_{r_i} and B_{r_e} such that S \subset \overline{B}_{r_e} \setminus B_{r_i} and r_e -r_i \leq C \, \mathrm {osc}(H) , with C depending only on n and upper bounds on the surface area of S and the C^2 regularity of S . Our approach is based on a quantitative study of the method of moving planes, and the quantitative estimate on r_e-r_i we obtain is optimal. As a consequence, we also prove that if osc (H) is small then S is diffeomorphic to a sphere, and give a quantitative bound which implies that S is C^1 -close to a sphere.

A proof of Alexandrov's uniqueness theorem for convex surfaces in [formula omitted

We give a new proof of a classical uniqueness theorem of Alexandrov [4] using the weak uniqueness continuation theorem of Bers–Nirenberg [8]. We prove a version of this theorem with the minimal regularity assumption: the spherical Hessians of the corresponding convex bodies as Radon measures are nonsingular.

A Generalization of Reilly's Formula and its Applications to a New Heintze–Karcher Type Inequality

In this paper, we prove a generalization of Reilly's formula in \cite{Reilly}. We apply such general Reilly's formula to give alternative proofs of the Alexandrov's Theorem and the Heintze-Karcher inequality in the hemisphere and in the hyperbolic space. Moreover, we use the general Reilly's formula to prove a new Heintze-Karcher inequality for Riemannian manifolds with boundary and sectional curvature bounded below.

Spinorial proofs of the Alexandrov Theorem for higher order mean curvatures in [formula omitted] and the Heintze–Karcher Inequality

We apply the key inequality (1) recently obtained by Hijazi and Montiel in [6] in the Euclidean space to some well-chosen spinor fields to obtain new proofs of the Alexandrov Theorem for higher order mean curvatures in Rn+1 (Theorem 1) and the Heintze–Karcher Inequality (Theorem 2).

Constant mean curvature surfaces in warped product manifolds

We consider surfaces with constant mean curvature in certain warped product manifolds. We show that any such surface is umbilic, provided that the warping factor satisfies certain structure conditions. This theorem can be viewed as a generalization of the classical Alexandrov theorem in Euclidean space. In particular, our results apply to the deSitter-Schwarzschild and Reissner-Nordstrom manifolds.

Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures

Given a positive function F on S n which satisfies a convexity condition, for 1 ≤ r ≤ n, we define the r-th anisotropic mean curvature function H F r for hypersurfaces in R n+1 which is a generalization of the usual r-th mean curvature function. We prove that a compact embedded hypersurface without boundary in R n+1 with H F r = constant is the Wulff shape, up to translations and homotheties. In case r = 1, our result is the anisotropic version of Alexandrov Theorem, which gives an affirmative answer to an open problem of F. Morgan.

Vector Measures on Topological Spaces

Abstract Let 𝑋 be a completely regular Hausdorff space, 𝐸 a quasi-complete locally convex space, 𝐶(𝑋) (resp. 𝐶𝑏(𝑋)) the space of all (resp. all, bounded), scalar-valued continuous functions on 𝑋, and 𝐵(𝑋) and 𝐵0(𝑋) be the classes of Borel and Baire subsets of 𝑋. We study the spaces 𝑀𝑡(𝑋,𝐸), 𝑀 τ (𝑋,𝐸), 𝑀 σ (𝑋,𝐸) of tight, τ-smooth, σ-smooth, 𝐸-valued Borel and Baire measures on 𝑋. Using strict topologies, we prove some measure representation theorems of linear operators between 𝐶𝑏(𝑋) and 𝐸 and then prove some convergence theorems about integrable functions. Also, the Alexandrov's theorem is extended to the vector case and a representation theorem about the order-bounded, scalar-valued, linear maps from 𝐶(𝑋) is generalized to the vector-valued linear maps.