[Paper Review] The Obata equation with Robin boundary condition

Xuezhang Chen, Mijia Lai+1 more|arXiv (Cornell University)|Jan 8, 2019

Nonlinear Partial Differential Equations8 references4 citations

TL;DR

This paper investigates the Obata equation with Robin boundary conditions on Riemannian manifolds, proving that non-constant solutions exist not only on spherical domains but also on other compact manifolds depending on the sign of the Robin parameter $ a $. The key result establishes a rigidity theorem showing that for $ a > 0 $, the manifold must be isometric to a spherical domain, with boundary components isometric to generalized Clifford tori or spheres, depending on the function's critical point structure and dimension.

ABSTRACT

We study the Obata equation with Robin boundary condition $\frac{\partial f}{\partial ν}+af=0$ on manifolds with boundary, where $a \in \mathbb{R}\setminus\{0\}$. Dirichlet and Neumann boundary conditions were previously studied by Reilly \cite{R}, Escobar \cite{Es} and Xia \cite{X}. Compared with their results, the sign of $a$ plays an important role here. The new discovery shows besides spherical domains, there are other manifolds for both $a>0$ and $a<0$. We also consider the Obata equation with non-vanishing Neumann condition $\frac{\partial f}{\partial ν}=1$.

Motivation & Objective

• To extend Obata's rigidity theorem to the case of Robin boundary conditions, generalizing prior results on Dirichlet and Neumann conditions.
• To determine the class of compact Riemannian manifolds admitting non-constant solutions to the Obata equation under Robin boundary conditions.
• To characterize the geometry of the boundary and the underlying manifold when the Robin parameter $ a $ is non-zero, particularly distinguishing between $ a > 0 $ and $ a < 0 $.
• To establish a connection between the number of critical points of the solution $ f $ on the boundary and the topological structure of the manifold, including covering maps and gluing constructions.
• To characterize the equality case in Lichnerowicz-type eigenvalue estimates under Robin boundary conditions, using the rigidity result.

Proposed method

• Analyzes the Obata equation $ \nabla^2 f + f g = 0 $ on compact manifolds with boundary under the Robin condition $ \frac{\partial f}{\partial \nu} + a f = 0 $, where $ a \neq 0 $.
• Uses normal geodesic coordinates near the boundary to derive asymptotic expansions of the metric and the function $ f $, expressing higher-order terms recursively.
• Applies the method of formal power series expansions in the radial variable $ r $, with coefficients determined by the initial data $ f_0, f_1 $ and the curvature terms.
• Derives recursive equations for the Taylor coefficients of $ f $ and the metric components $ \bar{g}_k $, using the structure of the connection coefficients and the curvature tensor.
• Establishes a gluing theorem for manifolds with matching Dirichlet and Neumann data, showing that two solutions with identical boundary data can be smoothly joined if one can be glued.
• Uses the fact that non-vanishing Neumann data on a dense subset ensures uniqueness of the asymptotic expansion, enabling smooth gluing.

Experimental results

Research questions

• RQ1 Which compact Riemannian manifolds admit non-constant solutions to the Obata equation under Robin boundary conditions?
• RQ2 How does the sign of the Robin parameter $ a $ affect the possible geometries of the manifold and its boundary?
• RQ3 Can the equality case in Lichnerowicz-type eigenvalue estimates for Robin boundary conditions be characterized via the Obata equation?
• RQ4 What is the role of the number of interior maximum points of $ f $ in determining the covering structure of the manifold?
• RQ5 Under what conditions can two manifolds with matching Dirichlet and Neumann data on the boundary be smoothly glued along the boundary?

Key findings

• For $ a > 0 $, any compact manifold $ (M,g) $ admitting a non-constant solution to the Obata equation with Robin condition must have constant sectional curvature $ 1 $, i.e., $ \mathrm{sec}_g \equiv 1 $.
• If $ f|_{\partial M} $ is constant, the boundary is connected and isometric to $ T^{n-1}(\theta) $, and $ (M,g) $ is isometric to a geodesic ball of radius $ \pi/2 - \theta $ in $ \mathbb{S}^n $.
• If $ f|_{\partial M} $ is not constant and all boundary components satisfy $ m(S) < n-2 $, then $ (M,g) $ is isometric to $ \mathbb{S}^n \setminus \bigsqcup_{i=1}^l D^{m_i}(\theta) $ with $ m_i < n-2 $.
• When some boundary component satisfies $ m(S) = n-2 $, the manifold is a $ k $-fold isometric covering of $ \mathbb{S}^n \setminus D^{n-2}(\theta) $, where $ k $ equals the number of interior maximum points of $ f $.
• For $ n = 2 $, each boundary curve is a closed curve of constant geodesic curvature $ -\cot\theta $, with length $ 2\pi k \sin\theta $, where $ k $ is the number of maximum points of $ f $ on that component.
• The equality case in Ren-Xu's Lichnerowicz-type eigenvalue estimate for Robin boundary conditions is characterized by the rigidity result: equality holds if and only if $ (M,g) $ is isometric to a spherical domain as above.