arXiv research

Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.

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

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.

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

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

Extends unique continuation theorem to manifolds with boundary, proving zero set codimension.

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.

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

Existence and uniqueness theorem for Ricci flow on weighted graphs proved.

Research proves unique continuation for Einstein-vacuum equations on aAdS spacetimes.

Study on moduli spaces of Seiberg-Witten equations on manifolds with boundary.

Researchers prove uniqueness and continuity of solution to L_p dual Minkowski problem.

Let D be a self-adjoint differential operator of Dirac type acting on sections in a vector bundle over a closed Riemannian manifold M. Let H be a closed D-invariant subspace of the Hilbert space of square integrable sections. Suppose D restricted to H is semibounded. We show that every element u in H has the weak uniqu…

SL(n) covariant valuations on Orlicz spaces are represented and characterized.

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 continuous geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes f…

Generalizes Candel's theorem on curvature of laminated surfaces.

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman …

The study proves the existence of $k$-convex hypersurfaces for specific curvature equations.

In this paper, we prove Matsushima's theorem for Kähler-Einstein metrics on a Fano manifold with cone singularities along a smooth divisor that is not necessarily proportional to the anti-canonical class. We then give an alternative proof of uniqueness of Kähler-Einstein cone metrics by the continuity method. Moreover,…

We study a second order differential equation corresponding to rotationally symmetric $F$-harmonic maps between certain noncompact manifolds. We show unique continuation and Liouville's type theorems for positive solutions. Asymptotic properties and the existence of bounded positive solutions are investigated.

We study a second order ordinary differential equation corresponding to rotationally symmetric $p$-harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive entire solutions. Asymptotic properties of the positive solutions are investiga…

We prove the two theorems of the title, settling two long standing questions in the local theory of singular minimal hypersurfaces. The sharpness of either result is with respect to its hypothesis on the size of the allowable singular sets. The proofs of both theorems rely heavily on the author's recent regularity and …

We prove that any properly oriented $C^{2,1}$ isometric immersion of a positively curved Riemannian surface M into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in M. A generalization of this result to nonnegatively curved surfaces is presented as well under suitable…

We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies…

Generalizing Riemannian theorems of Anderson-Herzlich and Biquard, we show that two $(n+1)$-dimensional stationary vacuum space-times (possibly with cosmological constant $Λ\in \R$) that coincide up to order one along a timelike hypersurface $\mycal T$ are isometric in a neighbourhood of $\mycal T$. We further prove th…

The purpose of this paper is to prove the uniqueness theorem of solutions of eigenvalue equations on one end of Riemannian manifolds for drift Laplacians, including the standard Laplacian as a special case; we shall impose "a sort of radiation condition" at infinity on solutions. We shall also provide several Riemannia…

Formalizes vNM utility theorem using Lean 4, proving existence and uniqueness.

Extends Killing vector fields to beyond compact Cauchy horizons.

New findings show non-uniqueness in Ricci flow solutions for dimensions n≥5.

Let $M$ be a closed connected spin manifold of dimension $2$ or $3$ with a fixed orientation and a fixed spin structure. We prove that for a generic Riemannian metric on $M$ the non-harmonic eigenspinors of the Dirac operator are nowhere zero. The proof is based on a transversality theorem and the unique continuation p…

Unique continuation result for expanding Ricci solitons.

Unique continuation property for measures in high dimensions.

We extend the Weil-Petersson metric to a projective variety with continuous local potentials.

Let $Ω$ be a smooth compact oriented 3-dimensional Riemannian manifold with boundary. A quaternion field is a pair $q=\{α,u\}$ of a function $α$ and a vector field $u$ on $Ω$. A field $q$ is {\it harmonic} if $α, u$ are continuous in $Ω$ and $\nablaα={\rm rot\,}u,\,{\rm div\,}u=0$ holds into $Ω$. The space ${\mathscr Q…

Unique solutions found for Plateau problems in smooth and continuous calibrations.

Defines conditions for unique conformal metrics on manifolds with specific curvature properties.

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…

Pathwise uniqueness shown for specific stochastic equations.

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

Research on unique continuation principles in medical and seismic imaging.

Let $(M,g)$ be a simple Riemannian manifold. Under the assumption that the metric $g$ is real-analytic, it is shown that if the geodesic ray transform of a function $f\in L^{2}(M)$ vanishes on an appropriate open set of geodesics, then $f=0$ on the set of points lying on these geodesics. The approach is based on a micr…

We provide precise formulations and proofs of two theorems from Darboux's lectures on orthogonal systems. These results provide local existence and uniqueness of solutions to certain types of first order PDE systems where each equation contains a single derivative for which it is solved: \[\frac{\partial u_i}{\partial …

Proves unique continuation for area minimizing currents.

We prove the unique continuation property at the conformal infinity for asymptotically hyperbolic Einstein metrics.

In this paper, we continue our study on a general time-inconsistent stochastic linear--quadratic (LQ) control problem originally formulated in [6]. We derive a necessary and sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional…

Extends curve theory to non-smooth data with finite curvature and torsion.

Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that th…

New guarantees for uniquely identifying transport maps and vector fields from finite measure-valued data.