We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index, the total curvature along the sequence is quantised in terms of the total curvature of some limit hypersurface, plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in Euclidean space -all of which are finite. Thus, we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary.
In this short article we investigate the topology of the moduli space of twoconvex embedded tori S n−1 × S 1 ⊂ R n+1 . We prove that for n ≥ 3 this moduli space is path-connected, and that for n = 2 the connected components of the moduli space are in bijective correspondence with the knot classes associated to the embeddings. Our proof uses a variant of mean curvature flow with surgery developed in our earlier article [3] where neck regions are deformed to tiny strings instead of being cut out completely, an approach which preserves the global topology, embeddedness, as well as two-convexity.equipped with the smooth topology, where Emb 2−conv ⊂ Emb denotes the space of smooth embeddings with the property that the sum of the smallest two principal curvatures is positive at every point. We proved that M 2−conv (S n ) is path-connected in every dimension n, and conjectured that M 2−conv (S n ) is actually contractible in every dimension n. This was inspired in part by Hatcher's proof of the Smale conjecture [6, 10] and by related work of Marques on the moduli space of metrics with positive scalar curvature [8].Here, we consider the moduli space of two-convex embedded tori, i.e. the spaceequipped with the smooth topology. Recall that by a result of Huisken-Sinestrari [7] the space M 2−conv (S n−j × S j ) is empty for 2 ≤ j ≤ n − 2, so without loss of generality we can assume j = 1 right away. A new interesting feature of the moduli space M 2−conv (S n−1 ×S 1 ) compared to M 2−conv (S n ), is that it can have non-trivial algebraic topology. In fact, some non-trivial algebraic topology can already be spotted at the level of π 0 , and our main theorem gives a complete classification of the path-components of M 2−conv (S n−1 × S 1 ).
We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient dimension less than eight and applicable to possibly improper limit hypersurfaces. In dimension three, this identity can be combined with the Gauss-Bonnet theorem to provide a constraint relating the topology of the free boundary minimal surfaces in a converging sequence, of their limit, and of the bubbles or half-bubbles that occur as blow-up models. We present various geometric applications of these tools, including a description of the behaviour of index one free boundary minimal surfaces inside a 3-manifold of non-negative scalar curvature and strictly mean convex boundary. In particular, in the case of compact, simply connected, strictly mean convex domains in R 3 unconditional convergence occurs for all topological types except the disk and the annulus, and in those cases the possible degenerations are classified.Mathematics Subject Classification (MSC 2010): Primary 53A10; Secondary 53C42, 49Q05.which holds true, with the setup as in Theorem 1, for all k sufficiently large. Actually, the second summand on the right-hand side can also be expressed only in terms of topological data (see Subsection 2.3 for a detailed discussion), so that we can derive the formula we will employ in all of our applications:Corollary 7. In the setting of Theorem 1, specified to n = 2, we have for all k sufficiently largewhere χ(Σ j ) denotes the Euler characteristic of Σ j and b j denotes the number of its ends. This is the starting point for our primary geometric applications. We present three instances, which are meant to illustrate the method, and leave other possible extensions in the form of remarks.Here is the first application we wish to discuss: since we can fully classify bubbles and half-bubbles of Morse index less than two (see Corollary 22 and Corollary 24) we can then get novel, unconditional, geometric convergence results for sequences of free boundary minimal
We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the singularities, we obtain a new Chern-Gauss-Bonnet formula with error terms that can be expressed as isoperimetric deficits. This is the first such formula in a dimension higher than two which allows the underlying manifold to have isolated branch points or conical singularities.
We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with large coupling constant.Comment: 29 pages, final version incorporating minor modifications based on comments from referees, to appear in J. London Math. So
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