Mean Field Equation of Liouville Type with Singular Data: Topological Degree

Abstract

We consider the following mean field equation: Δgv+ρ(h*ev∫Mh*ev−1)=4π∑j=1Nαj(δqj−1) on M, where M is a compact Riemann surface with volume 1, h* is a positive C1 function on M, and ρ and αj are constants satisfying αj > −1. In this paper, we derive the topological‐degree‐counting formula for noncritical values of ρ. We also give several applications of this formula, including existence of the curvature + 1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self‐gravitating strings. © 2015 Wiley Periodicals, Inc.

Metrics of constant positive curvature with four conic singularities on the sphere

“…This number has been found in some special cases, [23], [5], [7], [8], [9], [11], in particular, there is at most one such metric in the following two cases: a) when n ≤ 3 and the angles are not multiples of 2π, [23], [5], and b) when α j < 1 for all j, and n is arbitrary, [11]. However for large angles and n ≥ 4 there is usually more than one metric [3,7,8].…”

“…The ratio of two solutions f = w 1 /w 2 is a developing map of a metric in question if and only if the projective monodromy group of the Heun equation is conjugate to a subgroup of P SU(2) ≃ SO (3). In this case we say that the monodromy is unitarizable.…”

“…It follows from the results of Chen and Lin [3] that the number of metrics on a torus with one singularity with angle α is at least [(α − 1)/2] + 1 unless α is an odd integer.…”

Uniform bounds for solutions to elliptic problems on simply connected planar domains

“…It may be considered as a critical elliptic problem on planar domain, as the exponential nonlinearity is a natural counterpart of the Sobolev critical exponent in dimension greater or equal than 3. Solutions to (1.1) can been found either variationally ( [23,24,3,11]) or by computing the Leray-Schauder degree ( [12,13]), and blowing-up families have also been constructed ( [25,19]). In all of these cases the geometry and topology of the domain Ω play a fundamental role.…”

Existence and non existence results for the singular Nirenberg problem

“…We also refer the reader to [40], where the authors gave a criterion for the existence of a metric of constant curvature on S 2 . In a recent deep paper, [20], Chen and Lin computed the Leray-Schauder degree of (1.1) for λ / ∈ Γ(α m ) recovering some of the previous existence results and deriving new ones in the case χ(Σ) > 0. Anyway on the sphere there are still different situations in which the degree is zero and the solvability is not known.…”