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Your claim that "all solutions from general relativity are geodesically incomplete" is not true. The classical Schwarzschild/Kerr black hole solutions are geodesically incomplete, and along with Minko …

I'll add a few things which I've found helpful to get intuition to what Renato's already written. Scalar curvature has the very simple geometric interpretation as the volume defect of small balls (as …

I was going to post this as a comment, but it got too long. I'm not exactly sure how to answer the question as written, but the following might be enlightening: A crucial piece of information for your …

Below I sketch the proof of the following theorem: Theorem: Suppose that $\Sigma^n\subset (M^{n+1},g_0)$ is smooth and uniquely area-minimizing relative to it's boundary $\Gamma : = \partial\Sigma$ a …

Bochner's theorem extends to nonpositive Ricci to give: If $(M,g)$ is compact and has $\textrm{Ric}\leq 0$ then any Killing vector $X$ is parallel and $\textrm{Ric}(X,X) = 0$. See Petersen (3rd ed) …

Just a comment to supplement both nice answers. Robert Bryant alludes to the commonly held idea that since minimal surfaces are strictly stable on small scales (this is easy to prove) then the result …

This is false as stated. Take a surface of revolution generated by $(r(t),z(t))$. I claim I can choose the curve so that there are pieces that look like $(e^{-j},t)$ for $j$ large. The point is that i …

If you are OK with considering large balls, there are easy counterexamples. For example $T^2 \times \mathbb{R}$. Alternatively, there is a metric of negative Ricci curvature on $S^3$ (I think original …

Is there a smooth (noncompact) manifold $M$ such that for any Riemannian metric $g$ on $M$ there are $p_i$ and unit tangent vectors $v_i \in T_{p_i}M$ such that $Ricc(g)|_{p_i}(v_i,v_i) \leq -i$? This …

(i) This used to be a wide open area, but recently there has been some progress: Gabor Székelyhidi has constructed an example of an isolated singularity with a cylindrical tangent cone here: https://a …

I will assume that "converges in the critical set of $f$" is asking that if $f$ is MB then $\phi_t(y)$ (the flowlines) converge as $t\to\infty$ to $y_\infty$ a critical point (of course, depending on …

Is there a $C^\infty$-smooth embedding $\gamma : I \to \mathbb{R}^3$ so that there is no real analytic $2$-dimensional submanifold $M \subset \mathbb{R}^3$ with $\gamma(I)\subset M$?

You can construct a positively curved $(M^n,g)$ for any $n\geq 4$ that admits a stable minimal hypersurface. This is described in Example 1.2 here. (That paper also contains some non-existence results …

I think that this is not possible: Per my comment on Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics you want to solve $$ \textrm{div} (Y) = -2a\cdot x $$ for $Y$ or …

Positive scalar curvature implies that if $\textrm{index}(\Sigma_j)\leq I$ then $\Sigma_j$ have bounded area and genus. This is proven here https://arxiv.org/pdf/1509.06724.pdf (Theorem 1.3). That pap …