I will survey recent progress on the existence and regularity theory for harmonic maps from arbitrary closed manifolds to large classes of positively curved targets, with special emphasis on a natural family of sphere-valued harmonic maps which turns out to be intimately related to isoperimetric problems in spectral geometry, based on joint work with M. Karpukhin. In the case of two-dimensional domains, I will discuss applications of these techniques to the existence, regularity, and stability of metrics maximizing Laplace or Steklov eigenvalues on surfaces, highlighting some of the key ingredients in forthcoming work with Karpukhin, Kusner, and McGrath, in which these methods are employed to produce new families of minimal surfaces in $B^3$ and $S^3$ with prescribed topology.[-]

In the first part of this talk we will show how classical tools of Riemannian geometry can be used in the setting of stratfied spaces in order to obtain a lower bound for the spectrum of the Laplacian, under an appropriate assumption of positive curvature. Such assumption involves the Ricci tensor on the regular set and the angle along the stratum of codimension 2. We then show that a rigidity result holds when the lower bound for the spectrum is attained. These results, restricted to compact smooth manifolds, give a well-known theorem by M. Obata and A. Lichnerowicz.
Finally, we will explain some consequences of the previous theorems on the existence of a conformal metric with constant scalar curvature on a stratified space.[-]

I will survey recent progress on the existence and regularity theory for harmonic maps from arbitrary closed manifolds to large classes of positively curved targets, with special emphasis on a natural family of sphere-valued harmonic maps which turns out to be intimately related to isoperimetric problems in spectral geometry, based on joint work with M. Karpukhin. In the case of two-dimensional domains, I will discuss applications of these techniques to the existence, regularity, and stability of metrics maximizing Laplace or Steklov eigenvalues on surfaces, highlighting some of the key ingredients in forthcoming work with Karpukhin, Kusner, and McGrath, in which these methods are employed to produce new families of minimal surfaces in $B^3$ and $S^3$ with prescribed topology.[-]