arXiv research

Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.

In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqu…

Paper proves uniqueness of Type II Yamabe metrics on manifolds.

Proves uniqueness of capillary disks in 3D domains.

Uniqueness theorem for extremal charged black holes in de Sitter space.

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…

Paper proves uniqueness of minimal maps in curved spaces.

3D Schoenflies theorem for simply-connected 2-complexes.

Simplified proof of mass center system's uniqueness and generalized Pappus' theorem across Euclidean, spherical, and hyperbolic geometries.

We formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical theorem of Frobenius Theorem which says that an involutive C^1 distribution is uniquely integrable.

The paper proves uniqueness of Ricci flow on noncompact manifolds.

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a C^1 dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that …

Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete isometrically immersed submanifolds of ar…

We prove several unique continuation results for biharmonic maps between Riemannian manifolds.

Proves uniqueness of certain $S^1$-symmetric gravitational instantons.

Classical H.Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H.Minkowski uniqueness theorem due to A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be de…

We revisit the problem of uniqueness for the Ricci flow and give a short, direct proof, based on the consideration of a simple energy quantity, of Hamilton/Chen-Zhu's theorem on the uniqueness of complete solutions of uniformly bounded curvature. With a variation of this quantity and technique, we further prove a uniqu…

Revises Schwarzschild manifold rigidity proof for spin manifolds.

Develops a new framework for causal models on cyclic graphs, solving unique solvability issues.

Study proves existence and uniqueness of ancient flows from cones.

The paper proves unique characterization of gravitational instantons with specific volume growth.

For $φ$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $\int_X e^{-φ}.$ We prove that the logarithm of the volume is concave along bounded geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from…

Generalizes Thomas-Yau theorem for special and minimal Lagrangians.

In this paper, we prove a unique continuation or ``backwards-uniqueness'' theorem for solutions to the Ricci flow. A particular consequence is that the isometry group of a solution cannot expand within the lifetime of the solution.

Article strengthens initial data rigidity theorem to show unique spacetime extension.

Butscher, D. Lee, Y. Lee, and Joyce constructed a special Lagrangian submanifold by gluing a Lawlor neck into a transverse intersection point of two special Lagrangian submanifolds. We prove a uniqueness theorem for the gluing of flat special Lagrangian tori of real dimension 3 in a flat complex torus of complex dimens…

We consider $2$-dimensional integer rectifiable currents which are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by a suitable modification of White's original theorem for area minimizing currents…

New theorems prove uniqueness of solutions to geometric PDEs.

The paper proves existence and uniqueness of slant immersions in complex space forms.

In this note, we will show a backwards uniqueness theorem of the mean curvature flow with bounded second fundamental form in arbitrary codimension.

New static black hole uniqueness theorems for negative cosmological constant.

New theorem on graph curvature thresholds and uniqueness.

Ancient solutions to mean curvature flow have unique shapes.

The study proves the uniqueness of entropy-maximizing measures for geodesic flows on specific manifolds.

Study on uniqueness of hypersurfaces in hyperbolic space with constant mean curvature.

Proves uniqueness of small entropy self-expanders.

Proves monotonicity of parabolic frequency on all manifolds without curvature assumptions.

New theorem proves convex bodies with specific curvature measures are rescaled Wulff shapes.

In this paper, we study some basic geometric properties of pseudohermitian submanifolds of the Heisenberg groups. In particular, we obtain the uniqueness and existence theorems, and some rigidity theorems.

The singly periodic genus-one helicoid was in the origin of the discovery of the first example of a complete minimal surface with finite topology but infinite total curvature, the celebrated Hoffman-Karcher-Wei's genus one helicoid. The objective of this paper is to give a uniqueness theorem for the singly periodic gen…

We give an exposition of a theorem of Hirzebruch, Kodaira and Yau which proves the uniqueness of the Kahler structure of complex projective space, and of Yau's resolution of the Severi Conjecture.

We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remains open. Our main result states that, for every H>1/(\sqrt{3}), there exists a unique (up to left translations) immersed CM…

Study on PDEs in Heston model with unique solution and convergence proof.

Alternative proof for static black hole uniqueness with nonpositive mass.

The paper explores partial identifiability in nonnegative matrix factorization under specific conditions.

Proves new inequality for hyperbolic space hypersurfaces.

The proof of Theorem 7.12 of "Uniqueness of smooth cohomology theories" by the authors of this note is not correct. The said theorem identifies the flat part of a differential extension of a generalized cohomology theory E with ER/Z (there called "smooth extension"). In this note, we give a correct proof. Moreover, we …

Paper proves weak unique continuation for harmonic functions on RCD spaces but finds counterexample for strong uniqueness.