arXiv research

Uniformly elliptic Weingarten spheres in S2xR are congruent to a canonical example.

Study asymptotic behavior of Weingarten surfaces at infinity.

The study proves planes are the only complete uniformly elliptic Weingarten multigraphs.

We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+ε}, showing the opt…

The abstract manifold cannot have uniformly quasiregular self-maps.

In this paper we study the asymptotic behavior of second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence intro…

We prove that there is no nontrivial homogeneous order 2 solutions of fully nonlinear uniformly elliptic equations in dimension 4.

We show that for any $δ\in [0,1)$ there exists a homogeneous order $2-δ$ analytic outside zero solution to a uniformly elliptic Hessian equation in R^5.

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

Note on advancements in nonlinear elliptic equations' regularity theory.

We use the octonion algebra to construct singular solutions of Hessian fully nonlinear uniformly elliptic equations in 21 or more dimensions. The regularity of these solutions is the least possible one. The same is proven for Isaacs equtions.

New proof of Lorentzian splitting theorems using elliptic operators.

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 …

The paper studies gravitational instantons with flat limits and finds elliptic regularity estimates.

We show how to construct a non-smooth solution to Hessian fully nonlinear second-order uniformly elliptic equation using the Cartan isoparametric cubic in 5 dimensions.

We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is $C^{1,1}$. We obtain as a consequence a Liouville theorem for entire solutio…

We give sharp $C^{2,α}$ estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous results to complex Monge-Ampère equations with conical singularities. As an application…

A large class of variational equations for geometric objects is studied. The results imply conformal monotonicity and Liouville theorems for steady, polytropic, ideal flow, and the regularity of weak solutions to generalized Yang-Mills and Born-Infeld systems.

We prove that a Kleinian group $G$ acting upon $\mathbb{H}^{n}$ admits a non-constant $G$-automorphic function, even if it has torsion elements, provided that the orders of the elliptic (torsion) elements are uniformly bounded. This is accomplished by developing a technique for mashing distinct fat triangulations while…

For any elliptic K3 surface $\mathfrak{F}: \mathcal{K} \rightarrow \mathbb{P}^1$, we construct a family of collapsing Ricci-flat Kähler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov-Hausdorff limit to $\mathbb{P}^1$ equipped with the McLean metric. There are well-known e…

Paper proves linearity of solutions to degenerate elliptic equations in 3D.

Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.

The paper conjectures and proves fixed points for certain group actions on nonpositively curved spaces.

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations $F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold $M$. Finest results (with hypothesis that require the function $F$ to be degenerate ell…

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…

Copula is a powerful tool to model multivariate data. We propose the modelling of intraday financial returns of multiple assets through copula. The problem originates due to the asynchronous nature of intraday financial data. We propose a consistent estimator of the correlation coefficient in case of Elliptical copula …

The paper extends a Harnack inequality to noncompact evolving hypersurfaces.

In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds $X$. Precisely, we prove that a nonelementary discrete isometry subgroup of $\mathrm{Isom}(X)$ generated by two non-elliptic isometries $g$, $f$ contains a free subgroup of rank $2$ generated by isometries $f^N , h$ …

We construct non-symmetric diffusion processes associated with Dirichlet forms consisting of uniformly elliptic forms and derivation operators with killing terms on RCD spaces by aid of non-smooth differential structures introduced by Gigli '16. After constructing diffusions, we investigate conservativeness and the wea…

Calderón projector extended to fibred cusp operators.

For a sequence $\{(M_i, g_i, x_i)\}$ of pointed Riemannian manifolds with boundary, the sequence $\{(M_i,\tilde g_i,x_i)\}$ is its conformal satellite if the metric $\tilde g_i$ is conformal to $g_i$, that is, $\tilde g_i=u^{\frac{4}{n-2}}_ig_i$. Assuming the manifolds $(M_i,g_i,x_i)$ have uniformly bounded geometry, w…

In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products under appropriate geometric assumptions. As a consequence of this parametric study, we obtain very general uniqueness and non-existence results for a…

Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type m…

Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. Since then, a central problem in regularity theory has been to understand Schauder type estimates fashioning particular borderline scenarios. In such c…

Study shows limits of volume-constrained sets are finite unions of Wulff shapes.

Uniform elliptic theory for Dirac operators on orbifold resolutions.

In this paper we investigate the strict convexity and the differentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always differentiable in totally irrational dire…

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 …

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

This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on Q-rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior of a manifold with boundary, where the boundary has the structure of a tower of …

For any $n\ge 3$, $0<m\le (n-2)/n$, and constants $η>0$, $β>0$, $α$, satisfying $α\leβ(n-2)/m$, we prove the existence of radially symmetric solution of $\frac{n-1}{m}Δv^m+αv +βx\cdot\nabla v=0$, $v>0$, in $\R^n$, $v(0)=η$, without using the phase plane method. When $0<m<(n-2)/n$, $n\ge 3$, and $α=2β/(1-m)>0$, we prove…

Study on maximum principles for nonlinear equations on Riemannian manifolds.

We prove a Hardy inequality for uniformly elliptic operators subject to Dirichlet or mixed boundary conditions on domains $Ω$ with piecewiese smooth boundary in arbitrary Riemannian Manifolds (M, g). Employing an approach of E.B. Davies for the euclidean case, we show that it implies a sufficient geometric criterion un…

In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compac…

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…

This is an essay on potential theory for geometric plurisubharmonic functions. It begins with a given closed subset G of the Grassmann bundle $G(p,TX)$ of tangent $p$-planes to a riemannian manifold $X$. This determines a nonlinear partial differential equation which is convex but never uniformly elliptic (p < dim X). …

New flow defined to solve Hull-Strominger system, with estimates and convergence results.