Related papers: A Note on the Stability and Uniqueness for Solutio…
We generalize the following result of White: Suppose $N$ is a compact, strictly convex domain in $\RR^3$ with smooth boundary. Let $\Sigma$ be a compact 2-manifold with boundary. Then a generic smooth curve $\Gamma\cong \partial\Sigma$ in…
We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of…
In this paper, we study symmetry and existence of solutions of minimal gradient graph equations on punctured space $\mathbb R^n\setminus\{0\}$, which include the Monge-Amp\`ere equation, inverse harmonic Hessian equation and the special…
We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first (not necessarily linear) approximation of the given Poisson structure…
We study min-max theory for area functional among hypersurfaces constrained in a smooth manifold with boundary. A Schoen-Simon-type regularity result is proved for integral varifolds which satisfy a variational inequality and restrict to a…
In this paper, we study some existence and uniqueness results for systems of differential equations in which each of equations of the system involves a different Stieltjes derivative. Specifically, we show that this problems can only have…
We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior.…
Given a manifold with boundary, one can consider the space of subsurfaces of this manifold meeting the boundary in a prescribed fashion. It is known that these spaces of subsurfaces satisfy homological stability if the manifold has at least…
A pair of graphs $(\Gamma,\Sigma)$ is called unstable if their direct product $\Gamma\times\Sigma$ admits automorphisms not from $\mathrm{Aut}(\Gamma)\times\mathrm{Aut}(\Sigma)$, and such automorphisms are said to be unexpected. The…
We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to the complex plane in any complete orientable four-dimensional Riemannian manifold with uniformly positive isotropic curvature. We also…
We extend Newton's problem of minimal resistance to Riemannian surfaces endowed with a geodesic coordinate system, which includes the two-dimensional space forms such as the sphere and the hyperbolic plane. Assuming that the fluid particles…
In this paper, we present an analytical study, in the one space dimensional case, of the fluid dynamics system proposed in [4] to model the formation of biofilms. After showing the hyperbolicity of the system, we show that, in a open…
A `coherent system' $(\Cal E,V)$, consists of a holomorphic bundle plus a linear subspace of its space of holomorphic sections. Based on the usual notion in Geometric Invariant Theory, a notion of slope stability has been defined for such…
This is the second paper in a series studying the nonlinear stability of rarefaction waves in multi-dimensional gas dynamics. We construct initial data near singularities in the rarefaction wave region and, combined with the a priori energy…
We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that almost every such locally Hamiltonian flow with only simple saddles has singular…
For any $\Lambda>0$, let $\mathcal{M}_{n,\Lambda}$ denote the space containing all locally Lipschitz minimal graphs of dimension $n$ and of arbitrary codimension $m$ in Euclidean space $\mathbb{R}^{n+m}$ with uniformly bounded 2-dilation…
We present a theorem that allows to simplify linear stability analysis of periodic and quasiperiodic nonlinear regimes in N-particle mechanical systems (both conservative and dissipative) with different kinds of discrete symmetry. This…
In this article we extend several foundational results of the theory of complete minimal surfaces of finite index in the Euclidean space to minimal surfaces in asymptotically flat manifolds and, more generally, to marginally outer-trapped…
We extend Smale's singular bridge principle [Ann. of Math. 130 (1989), 603-642] for $n$-dimensional strictly stable minimal cones in $\mathbb{R}^{n+1}$ $(n \geq 7$) to arbitrary codimension and each $n \geq 3$. We then apply the procedure…
We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural…