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We prove that round balls of volume uniquely minimize in Gamow's liquid drop model. 

Abstract We prove Ilmanen’s resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface$$M^{n}\subset \mathbb{R}^{n+1}$$ $M^n\subset R^{n+1}$,$$2\leq n\leq 6$$ $2\le n\le 6$, with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens. 

Abstract We show that the mean curvature flow of generic closed surfaces in$$\mathbb{R}^{3}$$ $R^3$avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in$$\mathbb{R}^{4}$$ $R^4$is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons. 

We give a new proof of a recent result of Munteanu–Wang relating scalar curvature to volume growth on a $3$-manifold with non-negative Ricci curvature. Our proof relies on the theory of $\mathrm{\mu <\#comment/>}$-bubbles introduced by Gromov [Geom. Funct. Anal. 28 (2018), pp. 645–726] as well as the almost splitting theorem due to Cheeger–Colding [Ann. of Math. (2) 144 (1996), pp. 189–237]. 

Abstract The $$p$$ p -widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the $$p$$ p -widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets. We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the $$p$$ p -widths of the round sphere are attained by $$\lfloor \sqrt{p}\rfloor $$ ⌊ p ⌋ great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be $$\sqrt{\pi }$$ π . En route to calculating the $$p$$ p -widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero. 

Abstract We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $$\mathbf {R}^4$$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$ -close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular, this paper gives an alternative proof of our recent stable Bernstein theorem for minimal hypersurfaces in $$\mathbf {R}^4$$ . The new proof is more closely related to techniques from the study of strictly positive scalar curvature.