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Consider a “curve” defined by a list of points in finite dimension (here, four): pts = Table[{Cos[t], 0, Sin[2 t], Sin[t]}, {t, Subdivide[0, 1, 99]}] What would be a good approach to compute the local curvature? Several possibilities I thought of: interpolating pts and using ArcCurvature (introduced in Mathematica 10) using $ n+1$ consecutive pointsRead more

A famous theorem of Gromov says that the set of compact Riemannian manifolds with $ Ric \geq c$ and $ \text{diam} \leq D$ is relatively compact in the Gromov-Hausdorff metric. Chapter 10 of the book by Burago-Burago-Ivanov lists a few other such compactness theorems, but they are all dependent on some variant of global curvatureRead more

Disclaimer. I posted this question in Math.SE, but it haven’t received enough attention. Let $ \varphi_1$ be the first eigenfunction of the zero Dirichlet Laplacian in a planar bounded domain $ \Omega$ . That is, $ $ \left\{ \begin{aligned} -\Delta \varphi_1 &= \lambda_1 \varphi_1 &&\text{in } \Omega \subset \mathbb{R}^2,\ \varphi_1 &= 0 &&\text{on } \partialRead more

Is there a multi-valued fixed point theorem for complex projective Hilbert spaces, or equivalently, Kahler manifolds with positive curvature? In other words, an analogue of Kakutani’s theorem for complex projective Hilbert spaces. I am aware of Frankel’s theorem (https://msp.org/pjm/1961/11-1/pjm-v11-n1-p12-p.pdf) (theorem 4) which assures a fixed point for holomorphic functions from such a space to itself.Read more