An Obata singular theorem for stratified spaces

Abstract

Abstract. Consider a stratified space with a positive Ricci lower bound on the regular set and no cone angle larger than 2π. For such stratified space we know that the first non-zero eigenvalue of the Laplacian is larger than or equal to the dimension. We prove here an Obata rigidity result when the equality is attained: the lower bound of the spectrum is attained if and only if the stratified space is isometric to a spherical suspension. Moreover, we show that the diameter is at most equal to π, and it is equivalent for the diameter to be equal to π and for the first non-zero eigenvalue of the Laplacian to be equal to the dimension. We finally give a consequence of these results related to the Yamabe problem. Consider an Einstein stratified space without cone angles larger than 2π: if there is a metric conformal to the Einstein metric and with constant scalar curvature, then it is an Einstein metric as well. Furthermore, if its conformal factor is not a constant, then the space is isometric to a spherical suspension. MSC2010: 53 Differential geometry, 58 Global analysis, analysis on manifolds.

Non-existence of Yamabe minimizers on singular spheres

“…We can eventually apply Proposition 2.1 to u and obtain that it is a Lipschitz function. This, together with Corollary 4.8 of [24] allows us to prove the following:…”

“…The first statement (i) was proven in [24], Lemma 4.6. The integration by parts formula then follows easily by choosing a family of cut-off functions vanishing on an ε-tubular neighborhood Σ ε of the singular set and whose norm in L 2 tends to zero as ε goes to zero (see for example f ε in the proof of Theorem 2.1 in [23]).…”

“…Thanks to Theorem 4.1 in [24], we also know that if α x ≤ 2π for all x ∈ Σ n−2 , then an Einstein metric is also a Yamabe metric. This is not necessarily true when the cone angles along the singular set are greater than 2π (see Remark 1.4 for a simple example).…”

“…Our proof by contradiction depends on a combination of a branched covering version of Aubin's lemma for finite coverings [11], [7], the computation of the Yamabe constant of (S n , [h a ]) [22], [23], and regularity results for Yamabe minimizers [4], [22], [24]. We conjecture that an analog result should hold true for a ∈ (1, 2) as well.…”

Convergence of the Yamabe flow on singular spaces with positive Yamabe constant

“…We prove the theorem below. The assumptions on the Ricci curvature are made in order to apply the rigidity result of Mondello [Mon18] which yield the following.…”