arXiv research

New method for analyzing elliptic and parabolic equations.

Study proves uniform ellipticity implies uniform polyconvexity for anisotropic energy functionals.

Uniform K-homology theory applied to elliptic operators on manifolds with boundary.

We show that an elliptic uniform pseudodifferential operator over a manifold of bounded geometry defines a class in uniform K-homology, and that this class only depends on the principal symbol of the operator.

We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. The generalization will follow from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the uniform estimates …

Uniform estimates for elliptic problems near polygonal domains.

We consider non-elementary Kleinian groups Γ, without invariant plane, generated by an elliptic and a hyperbolic element with their axes lying in one plane. We find presentations and a complete list of orbifolds uniformized by such Γ.

Uniform elliptic theory for Dirac operators on orbifold resolutions.

Solves nonlinear problems on metric structures through eigenvalue counting.

We generalize Roe's Index Theorem for operators of Dirac type on open manifolds to elliptic pseudodifferential operators. To this end we introduce a class of pseudodifferential operators on manifolds of bounded geometry which is more general than similar classes defined by other authors. We revisit Spakula's uniform K-…

Solves modified Schouten tensor problems in conformal metric classes.

In this paper, we study Hessian equations and complex quotient equations on closed Hermitian manifolds. We directly derive the uniform estimate for the admissible solution. As an application, we solve general Hessian equations on closed Kähler manifolds.

Study fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.

We generalize Roe's index theorem for graded generalized Dirac operators on amenable manifolds to multigraded elliptic uniform pseudodifferential operators. This generalization will follow as a corollary from a local index theorem that is valid on any manifold of bounded geometry. This local formula incorporates the un…

We prove a uniform Sobolev inequality along the Sasaki-Ricci flow. In the process, we develop the theory of basic Lebesgue and Sobolev function spaces, and prove some general results about the decomposition of the heat kernel for a class of elliptic operators on a Sasaki manifold.

Constructs a new type of metric for elliptic surfaces.

Proves regularity of geodesic equation on Hermitian manifolds.

Researchers find a way to estimate potential functions for quaternionic metrics.

In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on Kähler manifolds. We first define a notion of weak solution of CSCK for an $L^\infty$ Kähler metric. The main result is to show that such a weak solution (with uniform $L^\infty$ bound…

The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.

We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of intrinsic elliptic Lagrangians which are sub-critical or critical. They are derived using uniform $ε-$regularity estimates whic…

Uniform estimates for complex equations on compact manifolds found.

We define a very general "parametric connect sum" construction which can be used to eliminate isolated conical singularities of Riemannian manifolds. We then show that various important analytic and elliptic estimates, formulated in terms of weighted Sobolev spaces, can be obtained independently of the parameters used …

In the present work we establish a quantization result for the angular part of the energy of solu- tions to elliptic linear systems of Schrödinger type with antisymmetric potentials in two dimension. This quantization is a consequence of uniform Lorentz-Wente type estimates in degenerating annuli. We derive from this a…

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem $(P, C)$ is equivalent to the uniform regularity of the natural family $(P_x, C_x)$ of associated boundary value …

In this paper, we study the existence and non-existence result of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics. We can get some useful local …

We study the parabolic complex Monge-Ampère type equations on closed Hermitian manfolds. We derive uniform $C^\infty$ {\em a priori} estimates for normalized solutions, and then prove the $C^\infty$ convergence. The result also yields a way to carry out method of continuity for elliptic Monge-Ampére type equations.

We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals consid…

We derive parametrizations of the Delaunay constant mean curvature surfaces of revolution that follow directly from parametrizations of the conics that generate these surfaces via the corresponding roulette. This uniform treatment exploits the natural geometry of the conic (parabolic, elliptic or hyperbolic) and leads …

The paper studies Harnack inequalities on Finsler metric measure spaces.

We present a somewhat new proof to the $C^{2,α}$-aprori estimate for the uniform elliptic Monge-Ampere equations, in both the real and complex settings. Our estimates do not need to differentiate the equation, and only depends on the $C^{α^{\prime}}-$norm of the right hand side of the equation, $0<α<α^{\prime}$.

Study Kähler metrics with constant scalar curvature using coupled equations.

We study the spherical cap packing problem with a probabilistic approach. Such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a stochastically independent uniformly distributed unit vector. When the set of unit vectors …

This paper is a self-contained presentation of certain aspects of the theory of weighted Sobolev spaces and elliptic operators on non-compact Riemannian manifolds. Specifically, we discuss (i) the standard and weighted Sobolev Embedding Theorems for general manifolds and (ii) Fredholm results for elliptic operators on …

Efficiently estimates covariance matrix for elliptical distributions under strong contamination.

PEA improves PCA and k-means for non-linear data and complex clusters.

Classifies self-shrinkers in arbitrary dimensions under specific curvature conditions.

Researchers solve metric curvature equations on manifolds with boundary.

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal …

Establishes interior regularity results for a broad class of two-dimensional nonlinear elliptic systems using a unified abstract framework.

Given a solution $u$ to a linear homogeneous second order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set $\Cr(u)\equiv \{x:|\nabla u|(x)=0\}$. The results are new even for harmonic functions on $\dR^n$. Given such a $u$, the standard {\it first o…

BGG-sequences offer a uniform construction for invariant differential operators for a large class of geometric structures called parabolic geometries. For locally flat geometries, the resulting sequences are complexes, but in general the compositions of the operators in such a sequence are nonzero. In this paper, we sh…

The study finds anisotropic minimal surfaces in 3-manifolds with smooth boundaries.

Unified kernel framework extends to stochastic systems, improving numerical stability.

Gursky-Streets introduced a formal Riemannian metric on the space of conformal metrics in a fixed conformal class of a compact Riemannian four-manifold in the context of the $σ_2$-Yamabe problem. The geodesic equation of Gursky-Streets' metric is a fully nonlinear degenerate elliptic equation and Gursky-Streets have pr…

We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature r…

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…

We consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the Ricci tensor to be a constant. It is related to the notion of minimal volumes in comparison geometry. Such a metric with the smallest eigenvalue of the Ricci tensor to be a constant is an extremal metric of volume in…