On some variational problems in geometry

Abstract

In this thesis we consider several variational problems in geometry that have a connection to the spectrum of the Laplacian acting on functions. In the first part, we study a quantity called the analytic systole, which was defined in recent joint work of the author with Werner Ballmann and Sugata Mondal. The motivation to study the analytic systole is its connection to small eigenvalues on Riemannian surfaces of finite topological type.We prove several qualitative and quantitative bounds for the analytic systole. The next few chapters deal with extremal metrics for Laplace eigenalues. Extremal metrics somewhat resemble the notion of critical points for the non-smooth functionals given by the eigenvalues of the Laplacian up to normalization. We study these functionals either on the space of all metrics with normalized volume or on a fixed conformal class. More precisely, we are interested in questions related to existence and regularity of extremal metrics. Firstly, we give an existence result for maximizers for the first eigenvalue on non-orientable surface relying on two spectral gap assumptions that prevent degenerations of a carefully chosen maximizing sequence in the moduli space. A similar spectral gap assumption occurs in a recent result of Petrides dealing with the orientable case. Slightly more general than actually required, these spectral gap assumptions ask, whether it is possible to strictly increase the first eigenvalue normalized by area by attaching a handle or a cross cap to a given closed Riemannian surface. We establish this under some extra assumptions. Unfortunately, our assumptions are too restrictive to establish the existence of maximizers at this point. However, there are some examples to which our techniques apply. In particular, we obtain the existence of a maximizing metric for the first eigenvalue on the surface of genus three Next, we consider extremal metrics for eigenvalues in a conformal class. Exploiting a connection of extremal metrics and n-harmonic maps we give an existence result for extremal metrics in perturbed conformal classes on products. In a similar way the connection to n-harmonic maps is used to prove a regularity result for extremal metrics. Finally, we exhibit a natural class of metrics with an integral scalar curvature bound in which one can maximize the first eigenvalue. More abstractly, we prove a regularity result for the Yamabe equation under an integral scalar curvature bound, provided the first eigenvalue is sufficiently large. In the last part, motivated by recent results on sharp eigenvalue bounds for the first eigenvalue on closed surfaces, we study the geometry of embedded, minimal surfaces of unbounded genus in ambient three-manifolds. Our main result here states that the systole of such a sequence tends to zero if the ambient manifold has positive Ricci curvature.