arXiv research

Proves well-posedness for elliptic problems on domains with singular points.

Research examines metrics with positive scalar curvature in domains with corners.

Proposes a regularization method for unsupervised domain adaptation that aligns predictions with target data's top singular vectors.

The paper classifies vertices in planar polygons formed by convex domains.

The study classifies area-maximizing hypersurfaces with singularities and exterior domains.

Study on images and singularities of pseudoholomorphic maps.

In this paper we give two examples of sequences of embedded minimal planar domains in $\mathbb{R}^3$ which converge to singular laminations of $\mathbb{R}^3$. In contrast with the situation for embedded minimal disks, these examples do not arise from complete embedded minimal planar domains and highlight some of the su…

Proves curvature comparison theorem for manifolds with conical singularities.

It is shown that most of the well-known basic results for Sobolev-Slobodeckii and Bessel potential spaces, known to hold on bounded smooth domains in $\mathbb{R}^n$, continue to be valid on a wide class of Riemannian manifolds with singularities and boundary, provided suitable weights, which reflect the nature of the s…

Equivalence proven between uniformizing varieties and tensors, generalizing uniformization results.

Defines renormalised energies for singular harmonic maps into compact manifolds.

The aim of this paper is to prove the existence of weak solutions to the equation $Δu + u^p = 0$ which are positive in a domain $Ω\subset {\Bbb R}^N$, vanish at the boundary, and have prescribed isolated singularities. The exponent $p$ is required to lie in the interval $(N/(N-2), (N+2)/(N-2))$. We also prove the exist…

Paper constructs two series of Lorentz bi-quotients from polyhedra.

The paper solves a generalized catenary problem in Lorentz-Minkowski space.

Paper extends Ohsawa-Takegoshi theorem to more general domains, proving removable singularities for plurisubharmonic functions.

Proves existence and uniqueness of solutions for a specific minimal surface equation.

Holomorphic family of strongly pseudoconvex domains in Kähler manifolds are studied.

The main result of this paper is a construction of fundamental domains for certain group actions on Lorentz manifolds of constant curvature. We consider the simply connected Lie group G~, the universal cover of the group SU(1,1) of orientation-preserving isometries of the hyperbolic plane. The Killing form on the Lie g…

Symplectic homology matches dual capacities for convex domains.

The study calculates best Sobolev constants with sharp Hardy terms in Euclidean and hyperbolic spaces.

MFSSA improves reconstruction accuracy of multivariate functional time series.

In this paper we prove the infinitesimal uniqueness theorem for the Newton potential of non simply connected bodies using the singularity theory approach. We consider the Newtonian potentials of the domains in ${\bf R}^n$ boundaries of which are the vanishing cycles on the level hypersurface of a holomorphic function w…

Research on refined algebraic domains respecting differential geometry.

New $G_2$-holonomy manifolds from 5d N=1 theories domain walls.

We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic $p \geq 2$. In this cas…

Study on moduli spaces of sextic curves with simple singularities and their compactifications.

In this article, by following the method in \cite{PT}, combining Willmore energy with isoperimetric inequalities, we construct two examples of singularities under mean curvature flow in $\mathbb{H}^3$. More precisely, there exists a torus, which must develop a singularity under MCF before the volume it encloses decreas…

In this work we will build a fundamental domain for Deligne-Mostow lattices in PU(2,1) with 2-fold symmetry, which complete the whole list of Deligne-Mostow lattices in dimension 2. These lattices were introduced by Deligne and Mostow using monodromy of hypergeometric functions and have been reinterpreted by Thurston a…

In this paper we extend our previous work on singularities of Monge-Ampère foliations to the case of pseudoconvex finite type domains. We are able to answer the questin of Burns on homogeneous polynomials whose logarithm satisfies the complex Monge-Ampère equation completely in dimension 2 . We are also able to general…

The paper generalizes curvature bounds for submanifolds with singularities.

The paper classifies energy-minimizing sets in specific domains.

Calderón projector extended to fibred cusp operators.

Using generalized Riemann maps, normal forms for almost complex domains (D, J) with singular foliations by stationary disks are defined. Such normal forms are used to construct counterexamples and to determine intrinsic conditions, under which the stationary disks are extremal disks for the Kobayashi metric or determin…

The study refines algebraic domains with specific boundary conditions.

We study the classification of singularities of holomorphic foliations and non-integrable one-forms under the hypothesis of transversality with real hypersurfaces.

New heat flow for harmonic maps avoids singularities but not bubbles.

It is a generally shared opinion that significant information about the topology of a bounded domain $Ω$ of a riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partialΩ}$, %, $d:Ω\rightarrow [0,\infty [$, from the boundary of $Ω$. To confirm such an idea we propose an approach based on the in…

We prove the existence of a continuous $BV$ minimizer with $C^{0}$ boundary value for the $p$-area (pseudohermitian or horizontal area) in a parabolically convex bounded domain. We extend the domain of the area functional from $BV$ functions to vector-valued measures. Our main purpose is to study the first and second v…

Study on conical singularities in 2D surfaces, deriving Polyakov formulas.

Energy minimizing harmonic maps between manifolds are known to be smooth outside a rectifiable set of codimension $3$, called the singular set. The possibility that this set is not a manifold, but has arbitrarily many small gaps in it, is not excluded in general. Here we prove that some part of the singular set - chara…

We consider the area preserving curve shortening flow with Neumann free boundary conditions outside of a convex domain or at a straight line. We give a criterion on initial curves that guarantees the appearance of a singularity in finite time. We prove that the singularity is of type II. Furthermore, if these initial c…

This paper develops a local analogue of the ADHM construction, which characterises ASD instantons defined over smooth bounded domains inside Euclidean $\mathbb{R}^4$ diffeomorphic to the 4-ball, in terms of infinite dimensional Hilbert spaces and bounded Hermitian linear operators satisfying an analogue of the ADHM equ…

Develops a new approach to study nonlinear PDEs and their singularities.

The first eigenfunction of a specific domain in hyperbolic space is log-concave.

In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy minimizing functions on $n$ dimensional domains and used it, in an essential way, to bound the Hausdorff dimension of the singular sets of area minimizing rectifiable currents of dimension $n$ and codimension $\geq 2$. Recent work of the second …

New theory of distributions on spaces with singular submanifolds.

Sharp rates and symmetry for higher order conformally invariant equations near singularities.

Researchers describe ECC of concave lens spaces and compute symplectic capacities.