Geometric flows and Riemannian geometry

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3. Other Geometric Flows

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Problem 3.1.
[B. Kotschwar, J. Streets] Must two Yang-Mills self-shrinkers that are asymptotic to the conical connection be actually identical?
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Problem 3.2.
[L. Ni]
Let $u:S^n \to \mathbb{R}$ be a positive function satisfying $u \cdot \det (D^2u+ug_{ij})=1,$ where $g_{ij}$ is the round metric on $S^n$ . Can one show that $u \equiv 1$ ?

More generally, if $u$ satisfies $u^{\frac{1}{\alpha}} \det (D^2u+ug_{ij})=1,$ for $\alpha > \frac{1}{n+2}$ , can one show that $u \equiv 1$ ?

The case $\alpha =\frac 1 n$ is known to be true. The case $n=1$ or $n=2$ with $\alpha \in (\frac 1 2, 1)$ is also true. Form the point of view of geometry, $u$ is the Gauss curvature.
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Problem 3.3.
[J. Streets] Let $S^2$ be the 2 dimension unit sphere and $g$ be a metric on $S^2$ with positive Gauss curvature. Take $g=e^{2u} ds^2$ . Consider the flow $\partial_t u=-1+\frac{\overline{K_u}}{K_u}$ where $\overline{K_u}$ is the average of $K_u$ . Known: long time existence; some weak convergence (a priori estimate degenerate at $t=\infty$ ).

Question: Does the flow converge to round? (If the genus is greater than 1, the analogous flow is understood). What about higher dimensions? More information for $n=2$ can be found on the paper on arxiv.

Cite this as: AimPL: Geometric flows and Riemannian geometry, available at http://aimpl.org/flowriemannian.