arXiv research

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…

Study end sum for surfaces and prove uniqueness results.

Paper explores non-uniqueness and uniqueness class for wave equations on graphs.

We prove some uniqueness result for solutions to the heat equation on Riemannian manifolds. In particular, we prove the uniqueness of $L^p$ solutions with $0< p< 1$, and improves the $L^1$ uniqueness result of P. Li by weakening the curvature assumption.

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

Revisits geometric PDE uniqueness in Riemannian and CR geometry.

Unique continuation result for expanding Ricci solitons.

Proves uniqueness of small entropy self-expanders.

Uniqueness of Kähler Ricci shrinkers proven on toric orbifolds.

Proves uniqueness of catenoid-like shapes in a ball.

The paper assesses conditions for the uniqueness of k-means clustering.

The study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.

Alternative proof for static black hole uniqueness with nonpositive mass.

Unique solution found for even L^p Minkowski problem at p=p0, but fails for p<p0.

Proves uniqueness of capillary disks in 3D domains.

The paper proves unique factorization of knotted handlebodies and examines handlebody-knot symmetry.

The paper proves uniqueness of evolving graphs by mean curvature flow under specific conditions.

The paper proves short-time existence and uniqueness of Ricci flow on Finsler manifolds.

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

We construct an example of a uniquely ergodic measured foliation on a surface such that the associated translation flow on the orientation double cover is minimal but not uniquely ergodic. We then prove a geometric criterion for the horizontal foliation of a quadratic differential to be uniquely ergodic. The second the…

New research proves uniqueness of maximal spacetime boundaries under certain conditions.

Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.

Unique solutions found for diffusive martingale problems.

The paper proves uniqueness of solutions to curvature problems using various methods.

Proves uniqueness of cylindrical tangent flows in mean curvature flow.

We prove some uniqueness results for positive harmonic functions on the unit ball satisfying a nonlinear boundary condition

Uniqueness proven for stable hypersurface tangent cones.

Paper proves unique tangent maps for complex maps into algebraic varieties.

Euler's elastica with monotone curvature is uniquely minimal.

The paper explores uniqueness and non-uniqueness of spacetime extensions in general relativity.

Proves uniqueness of barycenters on manifolds without restrictions.

The paper improves conditions for unique recovery in homomorphic sensing of subspaces.

Study existence and uniqueness of solutions for Yamabe problem on non-compact manifolds with negative curvature.

Study proves uniqueness of asymptotic limits for specific manifolds.

We prove uniqueness of solutions to complex Monge-Ampère equations for small temperature.

We show that Wang's proof of uniqueness of Anti-de Sitter spacetime can be adapted to provide uniqueness results for strictly static asymptotically locally hyperbolic vacuum metrics with toroidal infinity, and to prove negativity of the free energy $E-TS$ of asymptotically AdS black holes with higher-genus horizons.

The article recovers tensor fields from partial data using weighted divergent ray transforms.

Study on heat equation and eigenfunctions on RCD spaces, proving unique continuation.

Paper studies Kähler-Ricci flow convergence on Fano manifolds.

Manifolds uniquely identified by boundary distance differences.

We show uniqueness of classical solutions of the normalised two-dimensional Hamilton-Ricci flow on closed, smooth manifolds for smooth data among solutions satisfying (essentially) only a uniform bound for the Liouville energy and a natural space-time $L^2$-bound for the time derivative of the solution. The result is s…

Paper proves uniqueness of black holes and photon surfaces in higher dimensions.

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on, De Turck \cite{De} gave a simplified proof. In the later of 80's, Shi \cite{Sh1} generalized the local existence result to…

In a recent paper, the authors established the uniqueness of photon spheres in static vacuum asymptotically flat spacetimes by adapting Bunting and Masood-ul-Alam's proof of static vacuum black hole uniqueness. Here, we establish uniqueness of suitably defined sub-extremal photon spheres in static electro-vacuum asympt…

Let $E\to M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, ω)$. We prove that if $E$ admits a $ω$-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L. Biliotti and A. Ghigi implies the existence and uniqueness of $ω$-balanced metrics of certain dir…

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…

Unique K-polystable degenerations for Fano varieties confirmed.

In this paper, we give a height estimate for constant mean curvature graphs. Using this result we prove two results of uniqueness for the Dirichlet problem associated to the constant mean curvature equation on unbounded domains.