A novel type of Sobolev-Poincar\'e inequality for submanifolds of Euclidean space

For functions on generalised connected surfaces (of any dimensions) with boundary and mean curvature, we establish an oscillation estimate in which the mean curvature enters in a novel way. As application we prove an a priori estimate of the geodesic diameter of compact connected smooth immersions in terms of their boundary data and mean curvature. These results are developed in the framework of varifolds. For this purpose, we establish that the notion of indecomposability is the appropriate substitute for connectedness and that it has a strong regularising effect; we thus obtain a new natural class of varifolds to study. Finally, our development leads to a variety of questions that are of substance both in the smooth and the nonsmooth setting.Comment: 35 pages, no figure

An isoperimetric inequality for diffused surfaces

For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of varifold theory in the study of diffused surfaces.Comment: Awaiting publication in Kodai Math. J. The final printed version will be different. 14 pages, no figure

Properties of surfaces with spontaneous curvature

A model describing cell membranes as optimal shapes with regard to the $L^2$-deficit of their mean curvature to a given constant called spontaneous curvature is considered. It is shown that the corresponding energy functional is lower semi-continuous with respect to oriented varifold convergence on a space of surfaces whose second fundamental form is uniformly bounded in $L^2$. Elementary examples are presented showing that the latter condition is necessary. As a consequence, smoothly embedded minimisers among surfaces of higher genus are obtained. Moreover, it is shown that the diameter of a connected surface is controlled by the $L^1$-deficit of its mean curvature to the spontaneous curvature leading to an improved condition for the existence of minimisers. Finally, the diameter bound can be applied to obtain an isoperimetric inequality.Comment: 37 pages, no figure

Embedded Delaunay tori and their Willmore energy

A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below $8\pi$. Combining such a strict inequality with previous works by Keller-Mondino-Rivi\`ere and Mondino-Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution of the isoperimetric-constrained Willmore problem for tori. Similarly, we deduce the existence of smoothly embedded tori minimising the Helfrich functional with small spontaneous curvature. Moreover, it is shown that the tori degenerate in the moduli space which gives an application also to the conformally-constrained Willmore problem. Finally, because of their symmetry, the Delaunay tori can be used to construct spheres of high isoperimetric ratio, leading to an alternative proof of the known result for the genus zero case.Comment: 28 pages. Final version to appear in Nonlinear Analysis (TMA

A strict inequality for the minimisation of the Willmore functional under isoperimetric constraint

Inspired by previous work of Kusner and Bauer-Kuwert, we prove a strict inequality between the Willmore energies of two surfaces and their connected sum in the context of isoperimetric constraints. Building on previous work by Keller-Mondino-Rivi\`ere, our strict inequality leads to existence of minimisers for the isoperimetric constrained Willmore problem in every genus, provided the minimal energy lies strictly below $8\pi$. Besides the geometric interest, such a minimisation problem has been studied in the literature as a simplified model in the theory of lipid bilayer cell membranes.Comment: 16 pages. Final version to appear in Advances in Calculus of Variation

A priori bounds for geodesic diameter. Part II. Fine connectedness properties of varifolds

For varifolds whose first variation is representable by integration, we introduce the notion of indecomposability with respect to locally Lipschitzian real valued functions. Unlike indecomposability, this weaker connectedness property is inherited by varifolds associated with solutions to geometric variational problems phrased in terms of sets, $G$ chains, and immersions; yet it is strong enough for the subsequent deduction of substantial geometric consequences therefrom. Our present study is based on several further concepts for varifolds put forward in this paper: real valued functions of generalised bounded variation thereon, partitions thereof in general, partition thereof along a real valued generalised weakly differentiable function in particular, and local finiteness of decompositions.Comment: This is the second paper of a series of three papers which shall supersede arXiv:1709.05504 by the same authors. 58 pages (v1: 43 pages), no figures, added treatment of varifolds with free boundary which includes an isoperimetric lower density ratio bound in 9.22 and results in Theorem E and Corollary

The fiscal and intergenerational burdens of brakes and subsidies for energy prices

We study the effects of different financing rules for untargeted energy price brakes and subsidies on intergenerational welfare in a large-scale overlapping generations model. The results indicate that, in comparison with a laissez-faire solution without any government interventions, debt-financed implementations of such measures are very detrimental for young and future generations. However, the taxation of windfall profits can significantly contribute to reduce the economic burdens of these generations; whereas, the positive effects on older generations are much less pronounced

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Using Rauch’s comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li–Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon-type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch’s comparison theorem, the proofs mainly rely on the first variation formula and thus are valid for general varifolds