Related papers: An inverse problem for the Riemannian minimal surf…

We consider immersions of a Riemann surface into a manifold with $G_2$-holonomy and give criteria for them to be conformal and harmonic, in terms of an associated Gauss map.

It is known that every nonorientable surface $\Sigma$ has an orientable double cover $\tilde{\Sigma}$. The covering map induces an involution on the moduli space $\tilde{\M}$ of gauge equivalence classes of flat $G$-connections on…

We consider the inverse boundary value problem of the simultaneous determination of the coefficients $\sigma$ and $q$ of the equation $-\mbox{div}(\sigma \nabla u)+qu = 0$ from knowledge of the so-called Neumann-to-Dirichlet map, given…

Let $M$ be a compact Riemannian manifold not containing any totally geodesic surface. Our main result shows that then the area of any complete surface immersed into $M$ is bounded by a multiple of its extrinsic curvature energy, i.e. by a…

Let $\Omega\subset\r^n$ be a bounded mean convex domain. If $\alpha<0$, we prove the existence and uniqueness of classical solutions of the Dirichlet problem in $\Omega$ for the $\alpha$-singular minimal surface equation with arbitrary…

This article shows that for generic choice of Riemannian metric on a smooth manifold $M$ of dimension four, all prime compact parametrized minimal surfaces within $M$ have self-intersections in general position in the following sense:…

We prove the three embeddedness results as follows. $({\rm i})$ Let $\Gamma_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of…

Let $\Sigma$ be a $k$-dimensional minimal submanifold in the $n$-dimensional unit ball $B^n$ which passes through a point $y \in B^n$ and satisfies $\partial \Sigma \subset \partial B^n$. We show that the $k$-dimensional area of $\Sigma$ is…

A quadrangular embedding of a graph in a surface $\Sigma$, also known as a quadrangulation of $\Sigma$, is a cellular embedding in which every face is bounded by a $4$-cycle. A quadrangulation of $\Sigma$ is minimal if there is no…

In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the…

We consider inverse boundary value problems for elliptic equations of second order of determining coefficients by Dirichlet-to-Neumann map on subboundaries, that is, the mapping from Dirichlet data supported on $\partial\Omega\setminus…

We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N\leq -a^2\leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by $a$ and…

We study biminimal immersions, that is immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler-Lagrange equation associated to biminimal immersions for: i)…

Let $\Sigma$ be a hypersurface in an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We study the isometric extension problem for isometric immersions $f:\Sigma\to\mathbb R^n$, where $\mathbb R^n$ is equipped with the Euclidean…

We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which…

We obtain some estimates on the area of the boundary and on the volume of a certain free boundary hypersurface $\Sigma$ with nonpositive Yamabe invariant in a Riemannian $n$-manifold with bounds for the scalar curvature and the mean…

In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called \emph{simple…

In this note, we prove a Schwarz-Pick type lemma for minimal maps between negatively curved Riemannian surfaces. More precisely, we prove that if $f:M \to N$ is a minimal map with bounded Jacobian between two complete negatively curved…

We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body $\Omega\subset\mathbb{R}^{n}$ when the so--called Dirichlet-to-Neumann map is locally given on a non empty portion $\Gamma$ of the boundary…

A Real structure on a $4$-manifold $X$ is an orientation preserving smooth involution $\sigma$. We say that an embedded surface $\Sigma \subset X$ is Real if $\sigma$ maps $\Sigma$ to itself orientation reversingly. We prove that a…