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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. 

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. 

The Allen–Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang–Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show that for generic metrics on a 3-manifold, minimal surfaces arising from Allen–Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen–Cahn setting, a strong form of the multiplicity one-conjecture and the index lower bound conjecture of Marques–Neves in 3-dimensions regarding min-max constructions of minimal surfaces. Allen–Cahn min-max constructions were recently carried out by Guaraco and Gaspar–Guaraco. Our resolution of the multiplicity-one and the index lower bound conjectures shows that these constructions can be applied to give a new proof of Yau's conjecture on infinitely many minimal surfaces in a 3-manifold with a generic metric (recently proven by Irie–Marques–Neves) with new geometric conclusions. Namely, we prove that a 3-manifold with a generic metric contains, for every p = 1, 2, 3,…, a two-sided embedded minimal surface with Morse index p and area ~ p13, as conjectured by Marques-Neves.