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arXiv · 2311.11330

Generalized Ricci surfaces

Benoît Daniel·Yiming Zang
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Abstract

We consider smooth Riemannian surfaces whose curvature $K$ satisfies the relation $\Delta\log|K-c|=aK+b$ away from points where $K=c$ for some $(a,b,c)\in\mathbb{R}^3$, which we call generalized Ricci surfaces. We prove some isometric immersion theorems allowing points where $K=c$ using properties of log-harmonic functions. For instance, we obtain a characterization of Riemannian surfaces that locally admit minimal isometric immersions, possibly with umbilical points, into a $3$-dimensional Riemannian manifold of constant sectional curvature. We also give an application to convex affine spheres. Finally, we study compact generalized Ricci surfaces: we obtain topological obstructions and construct examples.

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Benoît Daniel, Yiming Zang. 2023-11-19. Generalized Ricci surfaces. https://arxiv.org/abs/2311.11330

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