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No, this is not true. For $L>0$, consider two lines in $\mathbb{R}^2$: $\{(x,0) : 0 \leq x \leq L\}$ and $\{(x,2) : 0 \leq x \leq L\}$. These can be "capped" at either end with half-circles of r …

I'm interpreting your question in the only reasonable manner that I can think of: $M= \mathbb{R}^3\backslash \{0\}$ and $g_m = \left( 1 + \frac{m}{2r}\right)\delta$ is the "doubled" Schwarzschild. …

A result of Fischer--Marsden (Duke "Deformations of the Scalar Curvature" 1975) says that if $(M,g)$ is scalar flat, but not Ricci flat, then the scalar curvature map $R :\{metrics\} \to \{functions\} …

The notation is potentially confusing, but the end result is correct. Essentially this claim is the relationship between the area and the energy of of a conformal map. Here is a slightly expanded pr …

My comment seemed to get jumbled, so here is an expanded version: In 3-d, an ancient complete Ricci flow (so, in particular a shrinking soliton) will have non-negative sectional curvature. This follo …

It isn't clear exactly what sort of initial conditions you're requiring. The difficulty with minimal initial conditions is part of the reason why it was an amazing result when Huisken--Ilmanen const …

Cheeger and Gromoll proved the well known splitting theorem: If $(M,g)$ is a complete manifold with $Ric\geq 0$ that contains a line, then $M$ splits isometrically as $M' \times \mathbb{R}$. He …

In general, asking whether or not all Jacobi fields on a minimal surface can be "integrated" to find a nearby minimal surface is a very difficult problem. For example, see Yau's remark here (page 246) …

The Yamabe flow is (up to a constant) the gradient flow of the Yamabe functional on the unit volume conformal class, as you expected. The comment by @Mark Peletier hints at your error: you aren't usin …

Consider the map $F:C^{2,\alpha}\to \mathbb{R}^A \times (C^{0,\alpha}\cap Im \Pi)$ $$ F: f \mapsto \left( \left(\int u_\infty^{\frac{4}{n-2}}(f-u_\infty)\psi_a\right)_{a\in A},\Pi\left( \frac{4(n-1)}{ …

A first answer is (under much weaker hypothesis than you require) Answer 1: If $\Sigma_k$ are a sequence of minimal surfaces with $\partial\Sigma_k \subset B(0,r_k)$ with $r_k\to\infty$ and $ …

I'm going to assume that you mean the following property $S \subset \mathbb{R}^3$ has property (*) if for any regular curve $\gamma\subset S$, the curvature vector $\vec{\kappa}$ of $\gamma$ as a …

I'm assuming you are supposing that $f\in C^\infty(M)$ for each $t \in [0,T]$, or something along this line. I'm not sure what happens if you allow $f$ to have some singularities. Then, picking any $0 …

One method is to repeat in dimensions $n$ Bryant's analysis in which he proves the existence and uniqueness of the so called Bryant soliton in $3$-dimensions: http://www.math.duke.edu/~bryant/3DRotSym …

This is a standard fact which is often asserted in the literature. A proof is given here: http://www.ugr.es/~jmmanzano/santalo/notes/GiuseppeTinaglia-Santalo-Granada.pdf (section 1), but I don't know …