Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature
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Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature Virginia Agostiniani1 · Mattia Fogagnolo2 · Lorenzo Mazzieri2
Received: 6 February 2019 / Accepted: 26 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension n ≥ 3. For every bounded open subset ⊂ M with smooth boundary, we prove that H n−1 n−1 , n − 1 dσ ≥ AVR(g) S
∂
where H is the mean curvature of ∂ and AVR(g) is the asymptotic volume ratio of (M, g). Moreover, the equality holds true if and only if (M\, g) is isometric to a truncated cone over ∂. An optimal version of Huisken’s Isoperimetric Inequality for 3-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue’s non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.
B Lorenzo Mazzieri
[email protected] Virginia Agostiniani [email protected] Mattia Fogagnolo [email protected]
1
Università degli Studi di Verona, strada Le Grazie 15, 37134 Verona, Italy
2
Università degli Studi di Trento, via Sommarive 14, 38123 Povo, TN, Italy
123
V. Agostiniani et al.
1 Introduction and main results The classical Willmore inequality [74] for a bounded domain of R3 with smooth boundary says that H 2 dσ ≥ 16π,
(1.1)
∂
where H is the mean curvature of ∂. Such an inequality has been extended in [19] to submanifolds of any co-dimension in Rn , for n ≥ 3. In particular, for a bounded domain in Rn with smooth boundary there holds H n−1 n−1 n − 1 dσ ≥ |S |,
(1.2)
∂
with equality attained if and only if is a ball. Implicit in this statement is the fact that the underlying metric by which H and dσ are computed is the Euclidean metric gRn . Note that the above rigidity statement can be rephrased by saying that the equality in (1.1) is fulfilled if and only if (∂, g∂ ) is homothetic to (Sn−1 , gSn−1 ), where g∂ is the metric induced by gRn on the submanifold ∂ and gSn−1 is the standard round metric. Recently, in [4], the Willmore-type inequality (1.2) and the corresponding rigidity statement have been deduced as a consequence of suitable monotonicity-rigidity properties of the function U (t) = t −(n−1)
|Du|n−1 dσ,
t ∈ (0, 1],
(1.3)
{u=t}
associated with the level set flow of the electrostatic potential u generated by the uniformly charged body . In other words, u is the unique harmonic function in Rn \ which vanishes at infinity and such that u = 1 on ∂. More precisely, what is proven in [4] is that the function U is nondecreasing and that this monotonicity is strict unless is a ball. Once this fact is established, the proof of (1.2) consists of a few lines. Indeed, exploiting first the global + feature of the monotonicity i.e.U (
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