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De Lellis-Topping inequalities on weighted manifolds with boundary. (English) Zbl 1491.53048

The De Lellis-Topping inequality implies that if a closed Riemannian manifold with nonnegative Ricci curvature is close to being an Einstein manifold, then its scalar curvature is close to being constant [C. De Lellis and P. M. Topping, Calc. Var. Partial Differ. Equ. 43, No. 3–4, 347–354 (2012; Zbl 1236.53036)]. In this paper, the authors prove a type of De Lellis-Topping inequality for a symmetric \((0,2)\)-tensor field \(T\) on a compact weighted Riemannian manifold with a convex boundary whose Bakry-Émery Ricci tensor is bounded from below by a negative constant. It is also assumed that the divergence of \(T\) is a constant multiple of \(\nabla(\operatorname{tr}T)\) and that \(T(\nu, -)\) is nonnegative on the boundary, where \(\nu\) is the outward unit normal.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
35P15 Estimates of eigenvalues in context of PDEs

Citations:

Zbl 1236.53036
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Full Text: DOI

References:

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