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Let $\Sigma$ be a compact surface and denote by $\nu$ the unit conormal for $\partial \Sigma$. Let $$E = \left\{ \phi \in C^{2,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\} $$ and $ …

Let $\mathcal{F}$ be a codimension one foliation in a closed $3$-manifold $M$. Does there exist an upper bound for the genus of the compact orientable leaves? That is, does there exist $G >0$ such tha …

Let $\{\Sigma_n\}_{n \geq 1}$ be a sequence of compact and orientable free boundary minimal surfaces embedded in $M$. …

This question has also been posted on MSE, but maybe here is the right place to post it. Is it true that if $D$ is a Riemannian $2$-disk having constant Gaussian curvature equal to $1$ and whose bound …

Let $(M^3,g)$ be a complete Riemannian manifold. Fix a two-sided immersion $\varphi : \Sigma^2 \to M$ from a closed surface into $M$, with unit normal $N$. Given $f \in C^\infty(\Sigma)$ and $\alpha : …

If this is not the case, what are the topological types of surfaces for which this supremum is finite? …

I asked this question on MSE, but obtained no answer. Maybe this is the right place to post it. Let $D$ be a properly embedded free boundary disk in the closed unit ball $\mathbb{B}^3$ of $\mathbb{R}^ …

Is it true that for a generic set of Riemannian metrics on $M$ the set of closed, connected and orientable embedded minimal surfaces of genus $g$ in $M$ is compact? …

Then, attaching small handles to $\Sigma$ would produce nonseparating surfaces of every genus $\geq g$. (sorry if it is a silly question…) …

My question is the following: if $(\Sigma, g)$ is any compact orientable Riemannian surface of genus $g \geq 1$, is it true that there are infinitely many closed, simple and geometrically distinct geo …

This is part of an argument in the paper “ On complete minimal surfaces with finite Morse index in three manifolds”, by Fischer-Colbrie. …