We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil3 and the Lie group Sol3 endowed with their standard left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil3 that lies on one side of some entire minimal graph G, then S is the image of G by a vertical translation. If S is a properly immersed minimal surface in Sol3 that lies on one side of a special plane Et (see the discussion just before Theorem 1.5 for the definition of a special plane in Sol3), then S is the special plane Eu for some u 2 R.

We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M, g) that are C0-asymptotic to Schwarzschild of mass m > 0. Refining an argument due to H. Bray, we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all sufficiently large volumes, and that they are close to centered coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volumepreserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.

Let k be the algebraic closure of a finite field of odd characteristic p, and X a smooth projective scheme over the Witt ring W (k) which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham flow1 and prove that the category of periodic Higgs-de Rham flows over X/W(k) is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the e ́tale fundamental group π_1(X_K ) of the generic fiber of X, after Fontaine-Laffaille and Faltings, where K is the fraction field of W(k). Moreover, we prove that every semistable Higgs bundle over the special fiber X_k of X of rank ≤ p initiates a semistable Higgs–de Rham flow and thus those of rank ≤ p − 1 with trivial Chern classes induce k-representations of π_1(X_K ). A fundamental construction in this paper is the inverse Cartier transform over a truncated Witt ring. In characteristic p, it was constructed by Ogus–Vologodsky in the nonabelian Hodge theory in positive characteristic; in the affine local case, our construction is related to the local Ogus–Vologodsky correspondence of Shiho.

Let$D$be a strictly pseudoconvex domain in$C$^{$n$}. We prove that $$\bar \partial u = \phi , \phi $$ , ϕ a $$\bar \partial $$ (0,1)-form, admits solutions in$L$^{$p$}(∂$D$), 1≤$p$<∞ and in BMO, under certain Wolff type conditions of ϕ. Some such results (for 1<$p$<∞) have previously been obtained by Amar in the ball, but under slightly stronger hypotheses. As a corollary we obtain a$H$^{$p$}-corona result for two generators.

This article is a continuation of our work on the classification of holomorphic framed vertex operator algebras of central charge 24. We show that a holomorphic framed VOA of central charge 24 is uniquely determined by the Lie algebra structure of its weight one subspace. As a consequence, we completely classify all holomorphic framed vertex operator algebras of central charge 24 and show that there exist exactly 56 such vertex operator algebras, up to isomorphism.