Abstract
We establish a new lower bound on the first nonzero eigenvalue \(\lambda _1 (\theta )\) of the sublaplacian \(\Delta _b\) on a compact strictly pseudoconvex CR manifold \(M\) carrying a contact form \(\theta \) whose Tanaka–Webster connection has pseudohermitian Ricci curvature bounded from below.
Highlights
The fundamental paper [15] by Sacks and Uhlenbeck approached the theory of harmonic maps from a Riemann surface M into a Riemannian manifold N, that is critical points u : M → N of the energy functional
The difficult part consists in controlling the limit α → 1. By studying this limit behavior of a sequence of α-harmonic maps as α 1, they obtained the existence of harmonic maps and insight into the formation of bubbles
Motivated by the supersymmetric nonlinear sigma model from quantum field theory, see [7], Dirac-harmonic maps from spin Riemann surfaces into Riemannian manifolds were introduced in [4]
Summary
The fundamental paper [15] by Sacks and Uhlenbeck approached the theory of harmonic maps from a Riemann surface M into a Riemannian manifold N , that is critical points u : M → N of the energy functional. Theorem 1.1 (Compactness and energy identity) Let (Mn, hn, cn, Sn) be a sequence of closed hyperbolic surfaces of genus g > 1 degenerating to a hyperbolic Riemann surface (M, h, c, S) by collapsing finitely many pairwise disjoint simple closed geodesics {γnj , j ∈ J }. Theorem 1.3 (Existence of Dirac-harmonic maps from degenerating surfaces) Let (Mn, hn, cn, Sn) be a sequence of closed hyperbolic surfaces of genus g > 1 degenerating to a hyperbolic Riemann surface (M, h, c, S) by collapsing finitely many pairwise disjoint simple closed geodesics {γnj , j ∈ J }. Corollary 1.6 Let (Mn, hn, cn, Sn) be a sequence of closed hyperbolic surfaces of genus g > 1 degenerating to a hyperbolic Riemann surface (M, h, c, S) with only Neveu-Schwarz type punctures by collapsing finitely many pairwisely disjoint simple closed geodesics {γnj , j ∈ J }.
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