We propose to study positive harmonic functions satisfying a nonlinear Neuman condition on a compact Riemannian manifold with nonnegative Ricci curvature and strictly convex boundary. A precise conjecture is formulated. We discuss its implications and present some partial results. Related questions are discussed for compact Riemannian manifolds with positive Ricci curvature and convex boundary.

中文翻译：

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具有凸边界和Ricci曲率从下界的紧黎曼流形。

我们建议在具有非负Ricci曲率和严格凸边界的紧黎曼流形上研究满足非线性诺曼条件的正谐波函数。提出了一个精确的猜想。我们讨论了其含义并提出了部分结果。讨论了具有正Ricci曲率和凸边界的紧黎曼流形的相关问题。