The Yamabe Constant on Noncompact Manifolds

Abstract

ABSTRACT. We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C 2 -topology, and that the Yamabe constant at infinity is even locally constant with respect to this topology. We also discuss to which extent the Yamabe constant is continuous with respect to coarser topologies on the space of Riemannian metrics.

Yamabe equation on some complete noncompact manifolds

“…It is worthy to point out that Y ∞ (M) > 0 implies that Y (M) > −∞ by Theorem 1.7 in [8]. Now we are ready to state our main results.…”

“…where ρ can be negative and α = α(ρ, n) > 0. 8 Proof. Given R > 1, first we fix a point x 0 ∈ M such that d(x 0 ) = 2R 2 , then we scale the metric by g = g/R 4 .…”

Stable solutions of the Yamabe equation on non-compact manifolds

“…The infimum is actually achieved. The minimizer is a solution of the Euler-Lagrange equation of the functional in parenthesis: (7) − n∆u + m ∇u By invariance if a function u is a minimizer so is cu λ given by cu λ (x) = cu(λx) for any constants c, λ ∈ R >0 . In terms of equation (6) this means that a solution u gives a 2-dimensional family of solutions.…”

“…The authors are supported supported by grant 220074 of CONACYT. study of Yamabe constants of noncompact manifolds can be found in [7]. See also [1,2,13] Our main motivation is to understand the Yamabe constants of certain non-compact Riemannian manifolds which play a central role in the study of the Yamabe invariants of closed manifolds (in particular when studying how the invariants behave under surgery, see [3]).…”

Untitled

“…In this case, there is a simple counterexample such that the Yamabe problem does not have a solution (see [Jin 1988]). See also [Aviles and McOwen 1988;Bland and Kalka 1989;Große and Nardmann 2014;Kim 1997;2000;Zhang 2003] and references therein for results related to the Yamabe problem on noncompact Riemannian manifolds. In particular, we mention the following result which is related to our main theorem.…”