On the Regularity of Alexandrov Surfaces with Curvature Bounded Below

Abstract

Abstract:In this note, we prove that on a surface with Alexandrov's curvature bounded below, the distance derives from a Riemannian metric whose components, for any p ∈ [ , ), locally belong to W ,p out of a discrete singular set. This result is based on Reshetnyak's work on the more general class of surfaces with bounded integral curvature.

On a singular Liouville-type equation and the Alexandrov isoperimetric inequlity

“…Let us observe that, by Theorem 4.4, e f +u = e ρ ∈ L p 0 ,loc loc (Ω \ S 2π ) ∩ L q 0 loc (Ω) for some p 0 > 2 and q 0 > 1, whence by standard elliptic estimates and the Sobolev embedding we find v n ∈ W 2,q 0 (Ω 1 ) ∩ C 0 (Ω 1 ). By using (3.26) with well known results in [24], we conclude that v n → u in W 1,r loc (Ω 1 ), for any r ∈ (1,2). At this point we observe that v n is a solution of,…”

“…Then, if (i) holds, we have u ∈ W 2,p,loc loc (Ω \ S 2π ) ∩ W 2,q loc (Ω) ∩ L ∞ loc (Ω), for some p > 2 and some q ∈ (1,2), and in particular u is a strong solution of (1.2), that is, it satisfies (1.7) with the equality sign. Moreover, in both cases, for any fixed simple and relatively compact subdomain E ⋐ Ω, we have M (E) < +∞ and the following inequality holds:…”

Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

“…Then, one could assume that Ric ≤ k to obtain a variant of Lemma 2.1 with the upper bound (log |∇u|) ≤ 2k. For this reason, we wonder whether there exists some class of assumptions implying the validity of (10).…”

“…Moreover, the BIC surfaces have a number of nice structure properties which permit to develop a differential calculus, including a singular (in some sense) Riemannian metric. We refer to [66,68,69] for comprehensive introductions to the theory of Alexandrov surfaces, see also the more recent [10,30,44] and references therein. A significant class of BIC surfaces consists of surfaces with conical singularities, i.e.…”

“…It turns out that d h,v is indeed a distance [69,Proposition 5.3]. In analogy with the smooth case, by using a common notation, we say that the distance d on S is induced by the (singular) Riemannian metric e 2v h; see also [10] for results about the regularity of this latter Riemannian metric. Since the harmonicity of a function is a conformal invariant in dimension two, the above relation permits to define harmonic functions on open subsets of (S, d) as the functions which are harmonic with respect to the metric h. More precisely, we define the Laplace-Beltrami operator of (S, d) by Recall that a surface (S, d) with bounded integral curvature is said to be without cusps, if ω({x}) < 2π for all x ∈ S. In particular, both the B I C surfaces with non-positive curvature measure and all the smooth surfaces are automatically without cusps.…”