In this paper, we investigate the regularity theory of $\textrm{codimension-}1$ integer rectifiable currents that (almost)-minimize parametric elliptic functionals. While in the non-parametric case it follows by De Giorgi–Nash’s Theorem that $C^{1,1}$ regularity of the integrand is enough to prove $C^{1,\alpha}$ regularity of minimizers, the present regularity theory for parametric functionals assume the integrand to be at least of class $C^2$. In this paper, we fill this gap by proving that $C^{1,1}$ regularity is enough to show that flat almost-minimizing currents are $C^{1,\alpha}$. As a corollary, we also show that the singular set has codimension greater than $2$.

Besides the result “per se”, of particular interest we believe to be the approach used here: instead of showing that the standard excess function decays geometrically around every point, we construct a new excess with respect to graphs minimizing the nonparametric functional, and we prove that if this excess is sufficiently small at some radius $R$ then it is identically zero at scale $R/2$. This implies that our current coincides with a minimizing graph there, hence it is of class $C^{1,\alpha}$.

We construct closed embedded minimal surfaces in the round three-sphere $\mathbb{S}^3 (1)$, resembling two parallel copies of the equatorial two-sphere $\mathbb{S}^2_{\mathrm{eq}}$, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of $\mathbb{S}^2_{\mathrm{eq}}$, or along the equatorial circle and the poles. To carry out these constructions we refine and reorganize the doubling methodology in ways which we expect to apply also to further constructions. In particular, we introduce what we call “linearized doubling”, which is an intermediate step where singular solutions to the linearized equation are constructed subject to appropriate linear and nonlinear conditions. Linearized doubling provides a systematic approach for dealing with the obstructions involved and also understanding in detail the regions further away from the catenoidal bridges.

We present a proof of the Riemannian Penrose inequality with charge in the context of asymptotically flat initial data sets for the Einstein–Maxwell equations, having possibly multiple black holes with no charged matter outside the horizon, and satisfying the relevant dominant energy condition. The proof is based on a generalization of Hubert Bray’s conformal flow of metrics adapted to this setting.

We show that a shrinking Ricci soliton with positive sectional curvature must be compact. This extends a result of Perelman in dimension three and improves a result of Naber in dimension four, respectively.

Consider an action of a connected compact Lie group on a compact complex manifold $M$, and two equivariant vector bundles $L$ and $E$ on $M$, with $L$ of rank $1$. The purpose of this paper is to establish holomorphic Morse inequalities à la Demailly for the invariant part of the Dolbeault cohomology of tensor powers of $L$ twisted by $E$. To do so, we define a moment map $\mu$ by the Kostant formula and we define the reduction of $M$ under a natural hypothesis on $\mu^{-1} (0)$. Our inequalities are given in term of the curvature of the bundle induced by $L$ on this reduction.