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. https://resolver.caltech.edu/CaltechTHESIS:04112011144739148
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 BowenMargulis 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 LedrappierPattersonSullivan measure is flip invariant, then c(x) is constant.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  19 April 1991 
Record Number:  CaltechTHESIS:04112011144739148 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:04112011144739148 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  6300 
Collection:  CaltechTHESIS 
Deposited By:  Tony Diaz 
Deposited On:  11 Apr 2011 23:31 
Last Modified:  02 Dec 2020 02:06 
Thesis Files

PDF
 Final Version
See Usage Policy. 2MB 
Repository Staff Only: item control page