Related papers: An inverse problem for general minimal surfaces

We construct a parabolic entire minimal graph $S$ over a finite topology complete Riemannian surface $\Sigma$ of curvature $-1$ and infinite area (thus of non-parabolic conformal type). The vertical projection of this graph yields a…

We give a short proof of the following fact. Let $\Sigma$ be a connected, finitely connected, noncompact manifold without boundary. If $g$ is a complete Riemannian metric on $\Sigma$ whose Gaussian curvature $K$ is nonnegative at infinity,…

This paper concerns the inverse spectral problem for analytic simple surfaces of revolution. By `simple' is meant that there is precisely one critical distance from the axis of revolution. Such surfaces have completely integrable geodesic…

In this paper we give results about projective embeddings of Riemann surfaces, smooth or nodal, which we apply to the inverse Dirichlet-to-Neumann problem and to the inversion of a Riemann-Klein theorem. To produce useful embeddings, we…

We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $\Gamma\subset \mathbb R^4$ such that the unique area minimizing surface spanned by $\Gamma$ has infinite…

Let N be a complete, homogeneously regular Riemannian manifold of dimension greater than 2 and let M be a compact submanifold of N. Let $\Sigma$ be a compact orientable surface with boundary. We show that for any continuous $f: (\Sigma,…

It is proved that the Heisenberg group $\operatorname*{Nil}\nolimits_{3}$ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product $\mathbb{T\times Z}$, where $\mathbb{T}$ is a totally geodesic…

Let $X$ be a smooth bordered surface in $\real^3$ with smooth boundary and $\hat \sigma$ a smooth anisotropic conductivity on $X$. If the genus of $X$ is given, then starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat \sigma}$ on…

This paper proposes direct and inverse results for the Dirichlet and Dirichlet to Neumann problems for complex curves with nodal type singularities. As an application, we give a method to reconstruct the conformal structure of a compact…

We prove that if a complete Riemannian surface $(\Sigma,d_\Sigma)$ is quasi-isometric to some bounded degree graph $G$, then $\Sigma$ admits a triangulation whose 1-skeleton is quasi-isometric to it when equipped with the simplicial metric.…

We consider the inverse problem of determining a potential in a semilinear elliptic equation from the knowledge of the Dirichlet-to-Neumann map. For bounded Euclidean domains we prove that the potential is uniquely determined by the…

The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is…

We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\it horizontal} area functional associated to the canonical…

We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.

Let $M$ be a compact Riemannian manifold of nonnegative Ricci curvature and $\Sigma$ a compact embedded 2-sided minimal hypersurface in $M$. It is proved that there is a dichotomy: If $\Sigma$ does not separate $M$ then $\Sigma$ is totally…

For a given simply connected Riemannian surface Sigma, we relate the problem of finding minimal isometric immersions of Sigma into S^2 x R or H^2 x R to a system of two partial differential equations on Sigma. We prove that a constant…

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…

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is…

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…

Given a complete Riemannian metric of nonnegative scalar curvature on $\Sigma \times (-\infty, 0 ] $, where $\Sigma$ denotes a $2$-sphere, we exhibit conditions that imply the existence of a closed minimal surface homologous to the…