arXiv research

Simple sphere eversion with a unique point.

Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries (spherical, Euclidean, hyperbolic, complex hyperbolic, to name a few) bisectors do not un…

Study shows unique tangent cones for area-minimizing currents at boundary points.

In this paper, we study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Mo…

Paper studies unique interior points and estimates for generalized translating soliton problems.

Study geodesic flows on hyperbolic manifolds without conjugate points, proving unique measure of maximal entropy.

We prove that tangent cones at singular boundary points of a two-dimensional current almost area minimizing are unique. Following the ideas exposed by White in [8], the result is achieved by combining a suitable epiperimetric inequality and an almost-monotonicity formula for the mass at boundary points.

Paper proves unique energy-minimizing curves in constrained spaces.

The paper investigates unique solutions for Fermat-Torricelli problem in specific norms.

The study restricts matrix group actions on CAT(0) spaces and uniquely arcwise connected spaces, proving fixed points are inevitable.

Proves unique continuation for area minimizing currents.

The paper studies harmonic maps between surfaces homotopic to a covering map, proving uniqueness and injectivity of Hopf differential.

Study shows non-uniqueness of Brakke flow near flat singular points.

We consider the issue of solution uniqueness for portfolio optimization problem and its inverse for asset returns with a finite number of possible scenarios. The risk is assessed by deviation measures introduced by [Rockafellar et al., Mathematical Programming, Ser. B, 108 (2006), pp. 515-540] instead of variance as in…

Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.

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

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…

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

In this paper we describe a method to establish when a symplectic manifold $M$ with semi-free Hamiltonian $S^{1}$-action is unique up to isomorphism (equivariant symplectomorphism). This will rely on a study of the symplectic topology of the reduced spaces. We prove that if the reduced spaces satisfy a rigidity conditi…

Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find \emph{all} fixed poi…

We prove the existence and uniqueness of harmonic maps in degree one homotopy classes of closed, orientable surfaces of positive genus, when the target has conic points with cone angles less than $2π$. For a cone point $p$ of cone angle less than or equal $π$ we show that one can minimize, uniquely, in the relative hom…

New height estimate for area minimizing currents, leading to unique tangent cones and decay properties.

Study on singularities in area-minimizing currents, proving unique tangent cones and rectifiability.

We introduce a class of hybrid marked point processes, which encompasses and extends continuous-time Markov chains and Hawkes processes. While this flexible class amalgamates such existing processes, it also contains novel processes with complex dynamics. These processes are defined implicitly via their intensity and a…

We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor …

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

For nearly spherical bodies, the unique center is proven under certain conditions.

Non-unique option pricing in Heston model analyzed mathematically.

Uniqueness proven for stable hypersurface tangent cones.

Analyzed Guyon's volatility model for existence and uniqueness.

We prove that for closed surfaces $M$ with Riemannian metrics without conjugate points and genus $\geq 2$ the geodesic flow on the unit tangent bundle $T^1M$ has a unique measure of maximal entropy. Furthermore, this measure is fully supported on $T^1M$ and the flow is mixing with respect to this measure. We formulate …

Geodesic flows on compact manifolds without conjugate points are shown to have a unique measure of maximal entropy.

Study proves uniqueness of tangent cones for area-minimizing currents in higher codimensions.

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

PUMA augments models to remove unique data points without performance loss.

In this article, we prove the existence of bounded solutions of quadratic backward SDEs with jumps, that is to say for which the generator has quadratic growth in the variables (z,u). From a technical point of view, we use a direct fixed point approach as in Tevzadze [38], which allows us to obtain existence and unique…

Study shows unique tangent flow for Lagrangian surfaces with bounded mean curvature.

It is known that for each combinatorial type of convex 3-dimensional polyhedra, there is a representative with edges tangent to the unit sphere. This representative is unique up to projective transformations that fix the unit sphere. We show that there is a unique representative (up to congruence) with edges tangent to…

Wave maps can have multiple bubbling solutions at blow-up points.

In this note, we combine the work of Ilmanen and of Colding-Ilmanen-Minicozzi to observe a uniqueness property for tangent flows at the first singular time of a smooth mean curvature flow of a closed surface in 3-dimensional Euclidean space. Specifically, if, at a fixed singular point, one tangent flow is a positive in…

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…

We show that for a closed surface of genus at least 5, or a surface of genus at least 2 with at least one marked point, the set of uniquely ergodic foliations and the set of cobounded foliations is path-connected and locally path-connected.

We introduce the gradient flow of the Seiberg-Witten functional on a compact, orientable Riemannian 4-manifold and show the global existence of a unique smooth solution to the flow. The flow converges uniquely in $C^\infty$ up to gauge to a critical point of the Seiberg-Witten functional.

Optimal geodesics connect boundary points in Teichmüller space.

We define and give explicit construction of the universal tree-graded space with a given collection of pieces. We apply that to proving uniqueness of asymptotic cones of relatively hyperbolic groups whose peripheral subgroups have unique asymptotic cones. Modulo the Continuum Hypothesis, we show that if an asymptotic c…

The paper identifies a component of representations mapping modular group elements to isometries with unique fixed points.

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

Unique tangent cones found for Kahler-Einstein metrics on singular varieties.