Session S33 - Spectral Geometry

Monday, July 12, 15:00 ~ 15:20 UTC-3

The Yamabe constants under constraints

Guillermo Henry

Universidad de Buenos Aires-IMAS CONICET, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Given a compact $n-$dimensional Riemannian manifold $(M,g)$ we say that $U$ is a solution of the Yamabe equation if satisfies (for some constant $c$) the equation $$\frac{4(n-1)}{(n-2)}\Delta_g U+s_gU=c|U|^{\frac{4}{n+2}}U, $$ where $s_g$ is the scalar curvature of $(M,g)$. From a geometric point of view study this equation is interesting because positive solutions are related to constant scalar curvature metrics in the conformal class of $g$. The Yamabe constant is the infimum of the total scalar curvature functional restricted to the conformal class. By the resolution of the celebrated Yamabe Problem this conformal invariant is achieved in any conformal class and it induces a positive solution of the Yamabe equation. On the other hand, the second Yamabe constant, which is the infimum of the second eigenvalues of the conformal Laplacian among the unit volume metrics conformal to $g$ is related to nodal solutions of the Yamabe equation, i.e., sign changing solutions.

In this talk we are going to discuss some results on the existence of solutions of the Yamabe equation obtained by considering the Yamabe constants under certain constraints like isoparametric functions or subgroups of the isometry group.

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