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In this Inventiones Mathematicae paper, Fischer-Colbrie proved the following result (Proposition 1): Proposition: Let $ M$ be a complete two-sided minimal surface in a three manifold $N$. Then if $M …

I am a bit confused about the statement of Theorem 1.1 in this paper by Brian White. For convenience, I will restate it here. Theorem: Let $\Omega$ be an open subset of a Riemannian $3$-manifold …

Let $\Sigma^n \subseteq \mathbb{R}^{n+1}$ be a minimal complete hypersurface. Let's consider $\bar{\Sigma} := \Sigma \times \mathbb{R}^k \subseteq \mathbb{R}^{n+k +1}$. Clearly $\bar{\Sigma}$ is a min …

Is there any Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$ in the literature? More precisely I would like to know if there is an answer to the following QUESTION: Let $f : \ma …

Colding and Minicozzi proved that any embedded minimal surface in $\mathbb{R}^3$ with finite topology must be proper and thus it can not be bounded. Is it possible to remove the assumption "finite t …

The well-known Half-space Theorem by Hoffman and Meeks says that there is no nonflat complete properly embedded minimal surface contained in an half space of $\mathbb{R}^3$. The higher dimensional v …

Let $\Omega \subseteq \mathbb{R}^2$ be a bounded convex domain with piecewise smooth boundary. Let $\phi :\partial \Omega \to \mathbb R$ be a continuous function. Let $L$ be a quasilinear elliptic o …

Let $(M, g)$ be a complete, noncompact Riemannian $n$-dimensional manifold and let $\phi \colon M \to \mathbb S^n$ be an harmonic map, where $\mathbb S^n$ is the euclidean $n$-dimensional sphere. Wh …

I would like to know if there exists a Liouville theorem for solutions $u : \mathbb{R}^n \to \mathbb{R}$ of uniformly elliptic equations of the kind $$ D_i \left( a_{ij} D_j u \right) + b_i D_i u = 0. …

Let $\Omega$ and open subset of $\mathbb{R}^n$. Let us consider the following operator: $$ \Delta_A (u)\, \, \colon= \text{div}(A \nabla u ), \qquad u \in C^{\infty}(\Omega) $$ where $A(x)$ is a matri …

In the Han and Lin book there is a Harnak inequality for elliptic operators of the kind: $$ L u = D_i \big(a^{ij}\, D_ju\big), $$ and the constant $C$ in the Harnack inequality does not depend on the …