John McCuan

Symmetry via spherical reflection and spanning drops in a wedge

We consider embedded ring-type surfaces (that is, compact, connected, orientable surfaces with two boundary components and Euler-Poincar e characteristic zero) in R 3 of constant mean curvature which meet planes 1 and 2 in constant contact angles 1 and 2 and bound, together with those planes, an open set in R 3 . If the planes are parallel, then it is known that any contact angles may be realized by innitely many such surfaces given explicitly in terms of elliptic integrals. If 1 meets 2 in an angle and if 1 + 2 > + , then portions of spheres provide (explicit) solutions. In the present work it is shown that if 1 + 2 + , then the problem admits no solution. The result contrasts with recent work of H.C. Wente who constructed, in the particular case 1 = 2 = =2 ,a self-intersecting surface spanning a wedge as described above. Our proof is based on an extension of the Alexandrov planar reflection procedure to a reflection about spheres [7], on the intrinsic geometry of the surface, and on a new maximum principle related to surface geometry. The method should be of interest also in connection with other problems arising in the global dierential geometry of surfaces.

Vertex theorems for capillary drops on support planes

We consider a capillary drop that contacts several planar bounding walls so as to produce singularities (vertices) in the boundary of its free surface. It is shown under various conditions that when the number of vertices is less than or equal to three, then the free surface must be a portion of a sphere. These results extend the classical theorem of H. Hopf on constant mean curvature immersions of the sphere. The conclusion of sphericity cannot be extended to more than three vertices, as we show by examples.

Symmetry via spherical reflection

In 1955 Alexandrov [2, 7, p. 147] introduced the method of planar reflection as a procedure for proving symmetry of closed surfaces of constant mean curvature. In subsequent years, a number of refinements were given, and the method was applied successfully to other problems in differential geometry and partial differential equations, cf. [10, 5, 12, 9, 3, 12]. In the present paper, we introduce a new procedure for obtaining symmetry which may be considered an extension of the planar reflection method to reflection about spheres.In the interest of clearly delineating the new features of the method, in this initial presentation we give a new proof of the original Alexandrov theorem: A closed embedded surface of constant mean curvature in Euclidean space is a round sphere. Our proof of this result is not simpler than that of Alexandrov; it is, however, different in some crucial respects.Our justification for introducing the new procedure lies in its applicability to configurations that are not accessible to planar reflection arguments. In particular, we use the new method in [8] to prove the non-existence in certain cases of embedded liquid bridges joining two planes that meet to form a wedge domain, a result whose interest derives in turn largely from the fact that under the same (physical) boundary conditions embedded bridges do exist and can be stable when the planes are parallel.The method of spherical reflection as presented below relies on a new maximum principle related to surface geometry, which we believe to have independent interest. Further, the method utilizes a peculiar characterization of spherical surfaces by aggregate symmetry, which should also have interest for other problems arising in the global differential geometry of surfaces.

Positively curved surfaces with no tangent support plane

We discuss a characterization of positively curved surfaces M with the property that, at each point, the tangent plane to M is not a support plane for the entire surface. Such positively curved surfaces with no tangent support plane necessarily have non-empty boundary, and any portion B C ∂M which has convex hull equal to the convex hull of ∂M we call a generating set. This set plays a key role in constructing examples. We give various examples among which there is an embedded topological disk with smallest possible generating set.

On Floating Equilibria in a Laterally Finite Container

The main contribution of this paper is the precise numerical identification of a model set of parameters for a floating object/container system which admits three distinct equilibrium configurations, two of which are local energy minimizers among pseudo-equilibrium configurations. This numerical result strongly suggests the existence of a physical system in which a circular object can be observed to float in a centrally symmetric position in two geometrically distinct configurations, i.e., at two different heights. Thus, the general dependence of observable stable equilibria on the physical parameters of the problem is both shown to be much more complicated than originally anticipated and likely to depend on additional information, e.g., the initial positioning of the floating object. We show the existence of at least one equilibrium configuration in any situation which the density of the floating object is less than that of the liquid bath. We also give a collection of conditions under which all equilibr...

New Geometric Estimates for Euler Elastica

Elastic curves, whose curvature depends linearly on height, were studied by L. Euler. We generalize an elementary comparison theorem relating pairs of these curves.

Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations

On dark matter, spiral galaxies, and the axioms of general relativity by H. L. Bray Embedded three-dimensional CR manifolds and the non-negativity of Paneitz operators by S. Chanillo, H.-L. Chiu, and P. Yang Aubry sets, Hamilton-Jacobi equations, and the Mane conjecture by A. Figalli and L. Rifford Minimal surfaces and eigenvalue problems by A. Fraser and R. Schoen On the maximal measure of sections of the

Liquid bridges, edge blobs, and Scherk-type capillary surfaces

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A variational formula for floating bodies

-cube by H. Konig and A. Koldobsky Extremities of stability for pendant drops by J. McCuan On the Funk-Radon-Helgason inversion method in integral geometry by B. Rubin Inequalities for the ADM-mass and capacity of asymptotically flat manifolds with minimal boundary by F. Schwartz Bounded extrinsic curvature of subsets of metric spaces by J. Wong