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Yamabe metrics on conical manifolds

Mattia Freguglia, Andrea Malchiodi

Abstract

We prove existence of Yamabe metrics on singular manifolds with conical points and conical links of Einstein type that include orbifold structures. We deal with metrics of generic type and derive a counterpart of Aubin's classical result. Interestingly, the singular nature of the metric determines a different condition on the dimension, compared to the regular case. We derive asymptotic expansions on the Yamabe quotient by adding a proper and implicit lower-order correction to standard bubbles, whose contribution to the expansion of the quotient can be determined combining the decomposition of symmetric two-tensor fields and Fourier analysis on the conical links.

Yamabe metrics on conical manifolds

Abstract

We prove existence of Yamabe metrics on singular manifolds with conical points and conical links of Einstein type that include orbifold structures. We deal with metrics of generic type and derive a counterpart of Aubin's classical result. Interestingly, the singular nature of the metric determines a different condition on the dimension, compared to the regular case. We derive asymptotic expansions on the Yamabe quotient by adding a proper and implicit lower-order correction to standard bubbles, whose contribution to the expansion of the quotient can be determined combining the decomposition of symmetric two-tensor fields and Fourier analysis on the conical links.
Paper Structure (10 sections, 23 theorems, 226 equations)

This paper contains 10 sections, 23 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Some preliminary facts
  3. Variations of geometric quantities and tensor decompositions
  4. Some computations in the conical setting
  5. Functional-analytic aspects and Yamabe quotient of cones
  6. Formal expansions of the Yamabe quotient
  7. Construction of the correction $\Psi$
  8. Role of the correction $\Psi$ in the expansion
  9. Expansion of the Yamabe quotient
  10. The four-dimensional case

Key Result

Theorem 1.1

Let $(M^n, \bar{g})$, $n \geq 4$, be compact with finitely-many conical points $\{P_1, \dots, P_l\}$ such that for all $i$ the metric $\bar{g}$ satisfies $(H_{P_i})$ with $\mathrm{Ric}(h_{i,0}) = (n-2) h_{i,0}$. Suppose there is a conical point $P$ whose link $(Y,h_0) \neq (S^{n-1}, g_{S^{n-1}})$ is Then $(M,\bar{g})$ admits a Yamabe metric.

Theorems & Definitions (50)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 40 more