Rigidity of three measure classes on the ideal boundary of mainifolds with negative curvature

Citation

Yue, Chengbo (1991) Rigidity of three measure classes on the ideal boundary of mainifolds with negative curvature. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4wkq-q530. https://resolver.caltech.edu/CaltechTHESIS:04112011-144739148

Abstract

On the ideal boundary, ∂M, of the universal covering of M of a negatively curved closed Riemannian manifold M, there exist three natural measure classes: the harmonic measure class {v_x}_(x∈M), the Lebesgue measure class {m_x}_(x∈M), the Bowen-Margulis measure class {u_x}_(x∈M). A famous conjecture (by A. Katok, F. Ledrappier, D. Sullivan) states that the coincidence of any two of these three measure classes implies that M is locally symmetric. We prove a weaker version of Sullivan’s conjecture: the horospheres in M have constant mean curvature if and only if m_x=v_x for all x ∈ M. In investigating these rigidity problems, we come across a class of integral formulas involving Laplacian Δ^u along the unstable foliation of the geodesic flow. One of which is ^∫_(SM) (Δ^u φ + < ∇^u log g, ∇^u φ >)dm = 0. Using these formulas, many rigidity problems are discussed, including (i) a simple proof of Hamenstädt’s lemma 5.3 which avoids her use of stochastic process, (ii) two functional descriptions of those manifolds which have horospheres with constant mean curvature: the horospheres in M have constant mean curvature if and only if ^∫_(SM) Δ^u φdm = 0 for all φ in C^2_u(SM) or ^∫_(SM) Δ^(su) φdm = 0 for all φ in C^2_(su)(SM). Finally, we study ergodic properties of Anosov foliations and their applications to manifolds of negative curvature. We obtain an integral formula for topological entropy in terms of Ricci and scalar curvature. We also show that the function c(x) in Margulis’s asymptotic formula c(x) = lim_(R→∞ e^(-hR)S(x,R) is almost always nonconstant. In dimension 2, c(x) is a constant function if and only if the manifold has constant negative curvature. Generally, if the Ledrappier-Patterson-Sullivan measure is flip invariant, then c(x) is constant.

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