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Results tagged with riemannian-geometry
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user 83841
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
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Upper bound of special Cheeger constant on $(S^2,g)$
$(S^2,g)$ is 2-dimensional sphere with Riemannian metric.The Cheeger constant of $(S^2,g)$ is
$$
h(S^2,g)=\inf_{\gamma} \frac{|\gamma|_g}{\min\{|A_1|_g, |A_2|_g\}}
$$
take the infimum over all closed …
- 165
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Low boundary of $\mathcal W$ function
Picture below is from Topping's Lectures on Ricci flow.
I don't understand the red line. From Lemma 8.1.8, I can get that $\mathcal W (g,f,\tau)$ has low boundary for any compatible $f,g,\tau$. But h …
- 165
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Using Rauch comparison theorem to get a comparison of two metric
Picture below is from Topping's Lectures on the Ricci flow. I've been stuck by the red line about two months. In fact, I asked it on ME two months ago. To describe the problem more precisely, conside …
- 165
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Cheeger constant and isoperimetric ratio
$(S^2,g)$ is 2-dimensional sphere with Riemannian metric. Consider any curves $\gamma$ on $S^2$ dividing the total area $A$ into two parts $A_1+A_2 =A$. The isoperimetric ratio is
$$
C_s(\gamma)=\frac …
- 165
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Principal eigenvalue of Laplacian under volume preserving mean curvature flow
Consider a compact uniformly convex n-dimensional hypersurface $M_0$ without boundary , which is smoothly imbedded in $\mathbb R^{n+1}$ , and suppose that $M_0$ is represented locally by some diffeomo …
- 165
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Derivation of the volume preserving mean curvature flow
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Picture above is from
Huisken, Gerhard, The volume preserv …
- 165