arXiv research

The paper solves curvature measure problem in hyperbolic space.

Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. In surprise, that methods has also brought us to promote Ivochkina's $C^2$ estimates for prescribed curvature equation in \cite{I1, I}.

Constructs polyhedral chains with prescribed tangent plane distributions.

We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$-convex bodies with prescribed $k$-th curvature measures ($k>0$) has been a longstanding problem. This is settled in this paper through the establishment of a crucial $C^2$ a priori estimate for the c…

The paper solves a problem of prescribing curvature measures on convex domains.

The paper extends entropy concepts to Monge-Ampère measures with prescribed singularities.

Study solves complex Hessian equations with prescribed singularities on compact Kähler manifolds.

We describe the space of measured foliations induced on a compact Riemann surface by meromorphic quadratic differentials. We prove that any such foliation is realized by a unique such differential $q$ if we prescribe, in addition, the principal parts of $\sqrt q$ at the poles. This generalizes a theorem of Hubbard and …

We consider a fully nonlinear partial differential equation associated to the intermediate $L^p$ Christoffel-Minkowski problem in the case $1<p<k+1$. We establish the existence of convex body with prescribed $k$-th even $p$-area measure on $\mathbb S^n$, under an appropriate assumption on the prescribed function. We co…

Study finds conditions for metrics on curved spaces.

Quantum states can be learned efficiently using gentle measurements.

We introduce new definitions of universal and superuniversal computable codes, which are based on a code's ability to approximate Kolmogorov complexity within the prescribed margin for all individual sequences from a given set. Such sets of sequences may be singled out almost surely with respect to certain probability …

A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole-singularities. In a neighborhood of a pole such a foliation comprises foliated strips and half-planes, and its leaf-space determines a metric graph. We introduce the notion of an asymptotic …

We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with conne…

Conditions for scalar curvature on compact manifolds under conformal deformation.

Study shows how to approximate functions using neural networks.

Study existence of achronal hypersurfaces with prescribed mean curvature in 3D spacetimes.

Paper proves existence and uniqueness of circle patterns on surfaces with assigned geodesic curvatures.

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures correspondin…

Paper solves a geometric problem involving mixtures of area and curvature measures.

Compactness results for hypersurfaces with mean curvature prescribed by an ambient function.

We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yam…

Solves curvature equations using parabolic flows in various spaces.

The paper studies strong topologies for complex Monge-Ampère equations on Kähler manifolds.

Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichmüller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichmüller space. Another way to state this…

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2π$, determines a holonomy representation …

Study on the convergence rate of prescribed scalar curvature flow.

Existence proved for $q$-Bass martingales with specific marginals.

Study of rays and co-rays in Wasserstein space, introducing Busemann functions.

Paper estimates diameter for Minkowski problem solutions.

The paper analyzes risk measures and optimal reserve allocation strategies.

The risk of financial positions is measured by the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate nondegeneracy, finiteness and continuity properties of these risk measures with respect to multiple eligible ass…

The Gordian distance between two knots measures how many crossing changes are needed to transform one knot into the other. It is known that there are always infinitely many non-equivalent knots `between' a pair of knots of Gordian distance two. In this paper we prove an extreme generalisation of this fact: there are kn…

Uniform systole bounds for arithmetic orbifolds and number fields.

Proves optimal regularity for sphere minimizers in 3-sphere.

The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\mathbb{R}^n$. A full solution to this is given when $1 < q < n$. The necessary and suffic…

Study classifies rotational hypersurfaces with prescribed mean curvature.

Develops variational approach for Kähler-Einstein metrics with prescribed singularities on Fano manifolds.

Proves existence and uniqueness of Killing graphs with prescribed curvature.

Paper estimates curvature of convex hypersurfaces with prescribed curvature.

Paper estimates curvature of semi-convex solutions in hyperbolic space.

We establish the Gaussian Double-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an $(n-2)$-dimensional plane at $120^{\circ}$ …

Theory proves existence of hypersurfaces with prescribed curvature.

Paper solves capillary Orlicz-Minkowski problem with new inequalities.

Solves Christoffel-Minkowski problem for capillary convex bodies in Euclidean half-space.

Solves Christoffel-Minkowski problem and Hessian equations with radial symmetry.

Introduce a flow for prescribed Hermitian-Yang-Mills tensors

We develop a compactness result near the boundary for families of locally convex immersions. We also develop a mod 2 degree theory for immersion of constant (and prescribed) Gaussian curvature with prescribed boundary. These are then used to solve the Plateau problem for immersions of constant (and prescribed) Gaussian…