arXiv subjects

John McCuan

Publications and source records attributed to John McCuan.

6 recordsLinked to original sources

arXiv1998

Rotations of the three-sphere and symmetry of the Clifford torus

We describe decomposition formulas for rotations of $R^3$ and $R^4$ that have special properties with respect to stereographic projection. We use the lower dimensional decomposition to analyze stereographic projections of great circles in $S^2 \subset R^3$. This analysis provides a pattern for our analysis of stereographic projections of the Clifford torus ${\mathcal C}\subset S^3 \subset R^4$. We use the higher dimensional decomposition to prove a symmetry assertion for stereographic projections of ${\mathcal C}$ which we believe we are the first to observe and which can be used to characterize the Clifford torus among embedded minimal tori in $S^3$---though this last assertion goes beyond the scope of this paper. An effort is made to intuitively motivate all necessary concepts including rotation, stereographic projection, and symmetry.

math.MG↗

arXiv1998

Isoperimetric flow and convexity of $H$-graphs

In this paper we consider a ``flow'' of nonparametric solutions of the volume constrained Plateau problem with respect to a convex planar curve. Existence and regularity is obtained from standard elliptic theory, and convexity results for small volumes are obtained as an immediate consequence. Finally, the regularity is applied to show a strong stability condition for all volumes considered. This condition, in turn, allows us to adapt an argument of Cabré and Chanillo which yields that any solution enclosing a non-zero volume has a unique nondegenerate critical point.

math.DG↗

arXiv1997

Embedded minimal ends asymptotic to the helicoid

The ends of a complete embedded minimal surface of {\em finite total curvature} are well understood (every such end is asymptotic to a catenoid or to a plane). We give a similar characterization for a large class of ends of {\em infinite total curvature}, showing that each such end is asymptotic to a helicoid. The result applies, in particular, to the genus one helicoid and implies that it is embedded outside of a compact set in ${\mathbb R}^3$.

math.DG↗

arXiv1997

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.

math.DG↗

arXiv1997

Retardation of Plateau-Rayleigh Instability: A Distinguishing Characteristic Among Perfectly Wetting Fluids

We consider a cylindrical film of fluid adhering to a rigid cylinder of fixed radius. The main result is to give the critical (maximum) length for which such a film of given thickness can be stable. The problem is considered both when the cylinder remains stationary and when the fluid and the cylinder are co-rotating at a fixed angular velocity.

math.AP↗

arXiv1995

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-Poincaré characteristic zero) in ${\bold R}^3$ of constant mean curvature which meet planes $Π_1$ and $Π_2$ in constant contact angles $γ_1$ and $γ_2$ and bound, together with those planes, an open set in ${\bold R}^3$. If the planes are parallel, then it is known that any contact angles may be realized by infinitely many such surfaces given explicitly in terms of elliptic integrals. If $Π_1$ meets $Π_2$ in an angle $α$ and if $γ_1+γ_2>π+α$, then portions of spheres provide (explicit) solutions. In the present work it is shown that if $γ_1+γ_2\leπ+α$, then the problem admits no solution. The result contrasts with recent work of H.C.~Wente who constructed, in the particular case $γ_1 = γ_2 =π/2$, a {\it 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, 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 differential geometry of surfaces.

math.DG↗