Related papers: Partial result of Yau's Conjecture of the first ei…

We prove partial regularity of stationary solutions and minimizers $u$ from a set $\Omega\subset \mathbb R^n$ to a Riemannian manifold $N$, for the functional $\int_\Omega F(x,u,|\nabla u|^2) dx$. The integrand $F$ is convex and satisfies…

Let $\Omega$ be a compact surface with smooth boundary and the geodesic curvature $k_g \ge {c > 0}$ along $\partial \Omega$ for some constant $c \in \mathbb{R}$. We prove that, if the Gaussian curvature satisfies $K \ge -\alpha$ for a…

Wang and Ye conjectured in [22]: Let $\Omega$ be a regular, bounded and convex domain in $\mathbb{R}^{2}$. There exists a finite constant $C({\Omega})>0$ such that \[ \int_{\Omega}e^{\frac{4\pi u^{2}}{H_{d}(u)}}dxdy\le C(\Omega),\;\;\forall…

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal…

In this paper we prove that if $u$ is a solution to second order hyperbolic equation $\partial^2_tu+a(x)\partial_tu-(div_x\left(A(x)\nabla_x u\right)+b(x)\cdot\nabla_x u+c(x)u)=0$ and $u$ is flat on a segment $\{x_0\}\times (-T,T)$ then $u$…

Given any asymptotically flat 3-manifold $(M,g)$ with smooth, non-empty, compact boundary $\Sigma$, the conformal conjecture states that for every $\delta>0$, there exists a metric $g' = u^4 g$, with $u$ a harmonic function, such that the…

Let \((M,\omega)\) be a compact symplectic manifold of dimension \(2n\) endowed with a Hamiltonian circle action with only isolated fixed points. Whenever \(M\) admits a toric \(1\)-skeleton \(\mathcal{S}\), which is a special collection of…

In this article, we give computable lower bounds for the first non-zero Steklov eigenvalue $\sigma_1$ of a compact connected 2-dimensional Riemannian manifold $M$ with several cylindrical boundary components. These estimates show how the…

In this paper, we prove a classification theorem for the stable compact minimal submanifolds of the Riemannian product of an $m_1$-dimensional ($m_1\geq3$) hypersurface $M_1$ in the Euclidean space and any Riemannian manifold $M_2$, when…

In this paper we give bounds for the first eigenvalue of the conformal Laplacian and the Yamabe invariant of a compact Riemannian manifold, by using conditions on the Ricci curvature and the diameter and deduce certain conditions on the…

We consider the higher order buckling eigenvalues of the following Dirichlet poly-Laplacian in the unit sphere $(-\Delta)^p u=\Lambda (-\Delta) u$ with order $p(\geq2)$. We obtain universal bounds on the $(k+1)$th eigenvalue in terms of the…

In this paper, we prove that the Chekanov-Eliashberg algebra of an horizontally displaceable n-dimensional Legendrian sphere in the contactisation of a Liouville manifold is a (n+1)-Calabi-Yau differential graded algebra. In particular it…

We consider rotated $n$-Laplace systems on the unit ball $B_1 \subset \mathbb{R}^n$ of the form \begin{align*} -\mathrm{div}\left( Q|\nabla u|^{n-2} \nabla u\right) = \mathrm{div}(G), \end{align*} where $u\in W^{1,n}(B_1;\mathbb{R}^N)$,…

We prove a weak version of the Perdomo Conjecture, namely, there is a positive constant $\delta(n)>0$ depending only on $n$ such that on any closed embedded, non-totally geodesic, minimal hypersurface $M^n$ in $\mathbb{S}^{n+1}$,…

This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and…

In this paper we study $N_d(k)$ the smallest positive integer such that any nice measure $\mu$ in $\R^d$ can be partitioned in $N_d(k)$ parts of equal measure so that every hyperplane avoids at least $k$ of them. A theorem of Yao and Yao…

We consider in this paper the nonlinear elliptic equation with Neumann boundary condition \begin{align*} \begin{cases} \Delta u=a|u|^{m-1}u\,\,\mbox{ in }\,\,\rnp\\ \dfrac{\partial u}{\partial t}=b|u|^{\eta-1}u+f\,\,\mbox{ on…

Sullivan's multi-bubble isoperimetric conjectures in $n$-dimensional Euclidean and spherical spaces assert that standard bubbles uniquely minimize total perimeter among all $q-1$ bubbles enclosing prescribed volume, for any $q \leq n+2$.…

In this paper, we consider smooth, properly immersed hypersurfaces evolving by mean curvature in some open subset of $\mathbb{R}^{n+1}$ on a time interval $(0, t_0)$. We prove that $p$ - integrability with $p\ge 2$ for the second…

In this paper, we verify the Glassey conjecture in the radial case for all spatial dimensions, which states that, for the nonlinear wave equations of the form $\Box u=|\nabla u|^p$, the critical exponent to admit global small solutions is…