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Mean curvature versus diameter and energy quantization Yasha Savelyev1 Received: 5 August 2019 / Accepted: 9 October 2019 © Fondation Carl-Herz and Springer Nature Switzerland AG 2019

Abstract We first partially extend a theorem of Topping, on the relation between mean curvature and intrinsic diameter, from immersed submanifolds of Rn to almost everywhere immersed, closed submanifolds of a compact Riemannian manifold. We use this to prove quantization of energy for pseudo-holomorphic closed curves, of all genus, in a compact locally conformally symplectic manifold. Keywords Energy quantization · Mean curvature · Intrinsic diameter Mathematics Subject Classification 53C42 Résumé Premièrement, nous étendons partiellement un théorème de Topping, concernant la relation entre la courbure moyenne et le diamètre intrinsèque, à partir des sous-variétés immergées de Rn aux sous-variétés presque partout immergées d’une variété Riemannienne compacte. Nous utilisons cette extension pour montrer la quantification de l’énergie pour les courbes fermées, pseudo-holomorphes, de tout genre dans une variété compacte, localement conformément symplectique.

1 Introduction For a compact almost complex manifold (M, J ) with metric g, energy quantization is a statement that there is a (M, g, J ) > 0 so that any non-constant pseudo-holomorphic map of a sphere into M has energy at least (M, g, J ). In order to carry out Gromov–Witten theory for locally conformally symplectic manifolds, or l. c. s. manifolds for short, we need in general a stronger form of energy quantization of pseudo-holomorphic curves, because it may be necessary to work with trivial homology classes (cf. [1]). In this case, if we are working with a higher genus curve, the energy of the curve a priori can collapse to 0, which would break compactness and invariance arguments.

Partially supported by PRODEP grant of SEP, Mexico.

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Yasha Savelyev [email protected] University of Colima, CUICBAS, Colima, Mexico

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Y. Savelyev

This is the “black earth” analogue of the “blue sky catastrophe” cf. [2–4]. We show that this cannot happen. The classical energy quantization is proved using mean value inequalities for pseudoholomorphic maps, [5,6]. To prove quantization of energy for higher genus curves in a l. c. s. manifold, we need more sophisticated differential geometry. In this note we first partially extend a theorem of Topping relating diameter and mean curvature of immersed submanifolds of Rn , and then use this to prove our quantization result.

1.1 Mean curvature vs diameter In [7] Topping gave via a concise but sophisticated argument, partially based on ideas of Ricci flows, a simple relation between intrinsic diameter and mean curvature, for immersed submanifold of Rn . Let us state it here: Theorem 1.1 [7] For  m a smoothly immersed closed submanifold of Rn we have:  diam() ≤ Const(m) |H|m−1 dvol, 

for H the mean curvature vector field along , vol the volume measure induced by the standard ambient metric, and diam the intrinsic diameter: max dist(,