Geometric analysis

Under Ricci flow $\partial_t g = -2\,\mathrm{Ric}(g)$, a round $n$-sphere of radius $r_0$ has $g(t) = (r_0^2 - 2(n-1)t)\,g_{S^n}$. At what time does a round 3-sphere of radius $r_0 = 4$ shrink to a point?

Which of these surfaces in $\mathbb{R}^3$ are minimal (mean curvature identically zero)? (Choose all that apply.)

Two coaxial rings of radius $1$ are placed at $x = \pm 0.5$. Two catenoids span them, with $a \approx 0.848$ (fat) and $a \approx 0.235$ (thin). Which statement is correct?

In the calibration proof that minimal graphs minimise area, $\omega(X, Y) = \det(\nu, X, Y)$ for the unit vector field $\nu$. Which property gives the inequality $\int_S \omega \le \mathrm{Area}(S)$ for every surface $S$?

A smooth embedded closed curve in the plane encloses area $4\pi$ and evolves by curve-shortening flow. At what time does it shrink to a point?

The Lean theorem round_sphere_ricci_flow proves, for $\rho(t) = r_0^2 - 2(n-1)t$ and $T = r_0^2/(2(n-1))$: $\rho' = -2(n-1)$, $\rho(0) = r_0^2$, $\rho(T) = 0$ and $\rho(t) > 0$ for $t < T$. What does it not establish?

For which compact targets $N$ does the Eells–Sampson theorem guarantee that every continuous map from a compact Riemannian manifold into $N$ is homotopic to a harmonic map? (Choose all that apply.)

Bernstein's theorem says an entire minimal graph over $\mathbb{R}^2$ is a plane. For which $n$ is every entire minimal graph over $\mathbb{R}^n$ a hyperplane?

Write heat_steps(u, lam, k): apply $k$ steps of the scheme $u_i \mapsto \lambda u_{i-1} + (1 - 2\lambda) u_i + \lambda u_{i+1}$ to the list u, with periodic indices (so u[-1] is the neighbour of u[0]). With $\lambda \le \tfrac12$ the maximum never increases; with $\lambda = 1$ it can overshoot.