arXiv research

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

New varifold solutions for mean curvature flow converge and are unique.

Study proves stability and uniqueness for a specific type of flow.

The paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.

Paper proves strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.

In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let $g(t)$ be a smooth complete solution to the Ricci flow on $\mathbb{R}^{3}$, with the canonical Euclidean metric $E$ as initial data, then $g(t)$ is trivial, i.e. $g(t)\equiv E$.

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

Paper studies Kähler-Ricci flow convergence on Fano manifolds.

The strong unique continuation property for Einstein metrics can be concluded from the well-known fact that Einstein metrics are analytic in geodesic normal coordinates. Here we give a proof of the same result that given two Einstein metrics with the same Ricci curvature on a fixed manifold, if they agree to infinite o…

Pathwise uniqueness shown for specific stochastic equations.

This part II of the paper is concerned with questions of existence and uniqueness of tangents in the special case of G-plurisubharmonic functions, where G is a compact subset of the Grassmannian of p-planes in ${\mathbb R}^n$. An upper semi-continuous function u on an open set $Ω$ in ${\mathbb R}^n$ is G-plurisubharmon…

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 …

This paper surveys some recent results on existence, uniqueness and removable singularities for fully nonlinear differential equations on manifolds. The discussion also treats restriction theorems and the strong Bellman principle.

The paper studies curves in Riemannian manifolds using total variation flow.

Novel weak solutions for volume-preserving mean curvature flow established.

The paper proves uniqueness of Ricci flow on noncompact manifolds.

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 …

Proves uniqueness of Ricci flow with scaling invariant estimates.

Proves existence and uniqueness of calibrated LSV model.

Study proves existence and uniqueness of ancient flows from cones.

We propose a weak formulation for the binormal curvature flow of curves in $\R^3.$ This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence t…

A minimal hypersurface in a sphere is uniquely determined.

Quantifies closeness of special Lagrangians under Floer conditions.

We derive sufficient conditions for the vanishing of plurigenera, $p_m(J), m>0$, on compact (l|k)-strong, $ω^l\wedge \partial\bar\partial ω^k=0$, Kaehler manifolds with torsion. In particular, we show that the plurigenera of compact (l|k)-strong manifolds, k<n-1, for which the holonomy of the unique Hermitian connectio…

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 strong existence and uniqueness, and Hölder regularity, of a large class of stochastic Volterra equations, with singular kernels and non-Lipschitz diffusion coefficient. Extending Yamada-Watanabe's theorem, our proof relies on an approximation of the process by a sequence of semimartingales with regularised ke…

We show the uniqueness of strictly convex closed smooth self-similar solutions to the $α$-Gauss curvature flow with $(1/n) < α< 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $α$-Gauss c…

Lecture notes on mean curvature flow for beginners.

In this short note, we prove that a bi-invariant Riemannian metric on $\mathrm{Sp}(n)$ is uniquely determined by the spectrum of its Laplace-Beltrami operator within the class of left-invariant metrics on $\mathrm{Sp}(n)$. In other words, on any of these compact simple Lie groups, every left-invariant metric which is n…

We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of usi…

In this note, we study submanifold geometry of the Atiyah-Hitchin manifold, the double cover of the $2$-monopole moduli space. When the manifold is naturally identified as the total space of a line bundle over $S^2$, the zero section is a distinguished minimal $2$-sphere of considerable interest. In particular, there h…

We provide a thorough analysis of the path-dependent volatility model introduced by Guyon \cite{G17}, proving existence and uniqueness of a strong solution, characterising its behaviour at boundary points, providing asymptotic closed-form option prices as well as deriving small-time behaviour estimates.

This is an expository paper giving a proof of the existence and uniqueness of smooth structures (hence also PL structures) on topological surfaces. Most published proofs rely on the topological Schoenflies theorem, but here we use instead the Kirby torus trick. This has the advantage of reducing the point-set topology …

With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities) allowing us to get rid of compactness assumptions on the manifold, and prove many exi…

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…

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

Study on geometric variational problems for existence, regularity, and uniqueness of solutions.

Researchers prove spectral uniqueness of complex/quaternionic structures on manifolds.

Paper proves uniqueness of catenary cylinders based on their asymptotic shape.

In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional $κ$-solutions. In this paper, we present an alternative proof for this fact and show that compact $κ$-solutions are rotational symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular…

Alexander trick applied to homology spheres for manifold homeomorphisms.

The subtle and unique imprint of dark matter substructure on extended arcs in strong lensing systems contains a wealth of information about the properties and distribution of dark matter on small scales and, consequently, about the underlying particle physics. However, teasing out this effect poses a significant challe…

We give partial answers to the following question: if $F$ is an $m$ by $m$ matrix on $\mathbb{R}^n$ satisfying a second order linear elliptic equation, does $\det F$ satisfy the strong unique continuation property? We give counterexamples in the case when the operator is a general non-diagonal operator and also for som…

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. …

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(κ)=σ\in(0, 1)$ with a prescribed asymptotic boundary $Γ$ at infinity has at least one smooth solution with uniformly bounded hyperbol…

Study develops numerical schemes for non-Markovian volatility models with memory.

Study collective pricing and hedging with admissible risk exchanges forming a finitely generated convex cone.

Study compares nodal sets of solutions to the Allen-Cahn equation.