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Let $(M,g)$ be a compact Riemannian manifold with boundary and assume it has positive scalar curvature. Question. Is it true that $DM$, the double of $M$, admits a metric of positive scalar curvature …

Let $M = (S^2 \times S^1) \setminus B$, where $B$ is a small open ball in $S^2 \times S^1$. Is it true to assert that every embedded $2$-disk $D \subset M$ such that $D \cap \partial M = \partial D$ n …

For $\varepsilon > 0$, we say that a closed, connected and oriented immersed hypersurface $M^n$ of a riemannian manifold $(N^{n+1},g)$ is $\varepsilon$-convex whenever all principal curvatures of $M$ …

Let $M^2$ be an orientable surface with boundary endowed with a Riemannian metric $g$. We know that if we put in the manifold $M \times \mathbb{R}$ the product metric $\overline{g} = g + dt^2$, then t …

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 $M^3$ be a closed orientable smooth manifold, let $g_n$ be a sequence of Riemannian metrics on $M$ converging to $g$ and let $\Sigma_n$ be a sequence of closed orientable $g_n$-minimal surfaces em …

I posted this question on MSE and I would like to see if my reasoning is correct. Let $M^3$ be a compact, connected and oriented $3$-manifold with nonempty boundary and let $\Sigma^2$ be a compact and …

I am sorry if this is a basic question, but I don't think in MSE I will receive any answers. Let $(M^3,g)$ be a compact and oriented Riemannian $3$-manifold. Let $\alpha$ and $\beta$ be the integral c …

Let $(M^3,g)$ be a compact Riemannian $3$-manifold with boundary. In a paper by L. Ambrozio, he considers the set $\mathcal{F}_M$ of all immersed disks in $M$ whose boundaries are curves in $\partial …

Let $\Gamma$ be a nontrivial group of isometries of $\mathbb{S}^n$, $n \geq 2$, acting properly discontinuously. For $p \in \mathbb{S}^n$, define $$r(p) = \min_{g \in \Gamma \setminus\{e\} } d(p, g(p …

Consider the following theorem, proved in this paper: Theorem (Theorem 6.1). Suppose we have a sequence $(\Sigma_j, \partial \Sigma_j) \subset (M, \partial M)$ of immersed free boundary minimal …

Let $(M^3,g)$ be a compact, connected and oriented Riemannian $3$-manifold with boundary. For a harmonic map $u : M \to \mathbb{S}^1$ satisfying Neumann condition along $\partial M$, let $h = u^*(d \t …

This question has been posted on MSE with no answers. Let $M^3$ be a closed, connected and orientable smooth $3$-manifold and let $\mathring{M}$ denote the manifold $M$ with $n$ disjoint open balls $B …

This question is related to a previous one. Let $(M^n,g)$ be a compact Riemannian manifold with boundary. Assume it has positive scalar curvature and $\partial M$ is mean convex (positive mean curvatu …

I also posted this question on MSE, but since it may be a delicate question, I decided to post it here. Given a Riemannian manifold $(M^n,g)$ and an integer $1 \leq k \leq n-1$, the $k$-systole of $M$ …