Stability of triply periodic minimal surfaces

In 1992, Ross proved that some classical triply periodic minimal surfaces in three-dimensional Euclidean space (Schwarz P surface, D surface, and Schoen's gyroid) are stable for volume-preserving variations. This paper extends the result to four one-parameter families of triply periodic minimal surfaces, namely, tP family, tD family, rPD family, and H family. We obtain sufficient conditions for volume-preserving stability, and as their numerical applications, we prove that, for each family, every triply periodic minimal surface with Morse index one is volume-preserving stable.     

The first Steklov eigenvalue, conformal geometry, and minimal surfaces

We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue σ₁ of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Σ with genus γ and k boundary components we obtain the upper bound σ₁L(∂Σ)?2(γ+k)π. For γ=0 and k=1 this result was obtained by Weinstock in 1954, and is sharp. We attempt to find the best constant in this inequality for annular surfaces (γ=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that σ₁(Σ)L(∂Σ) is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. Motivated by the annulus case, we show that a proper submanifold of the ball is immersed by Steklov eigenfunctions if and only if it is a free boundary solution. We then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least π, and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two-dimensional case.     

Laguerre minimal surfaces,isotropic geometry and linear elasticity

Laguerre minimal (L-minimal) surfaces are the minimizers of the energy \(\int (H^2-K)/K d\!A\). They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of \(\int (H^2-K)d\!A\), which are known to be an entity of Möbius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, L-minimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between L-minimal surfaces of the spherical type, isotropic minimal surfaces (graphs of harmonic functions), and Euclidean minimal surfaces. This relation exhibits connections to geometrical optics. In this paper we also address and illustrate the computation of L-minimal surfaces via thin plate splines and numerical solutions of biharmonic equations. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to L-minimal surfaces and certain Lie transforms of L-minimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.     

Triply periodic minimal surfaces bounded by vertical symmetry planes

We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz–Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces. S. Fujimori was partially supported by JSPS Grant-in-Aid for Young Scientists (Start-up) 19840035. M. Weber’s material is based upon work for the NSF under Award No. DMS-0139476.     

The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends

Given an integer , let be the space of complete embedded singly periodic minimal surfaces in , which in the quotient have genus zero and Scherk-type ends. Surfaces in can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is well known. It is also known that consists of the -parameter family of singly periodic Scherk minimal surfaces. We prove that for each , there exists a natural one-to-one correspondence between and the space of convex unitary nonspecial polygons through the map which assigns to each the polygon whose edges are the flux vectors at the ends of (a special polygon is a parallelogram with two sides of length and two sides of length ). As consequence, reduces to the saddle towers constructed by Karcher.

     

Some existence and uniqueness theorems for doubly periodic minimal surfaces

Oblatum 5-XI-1991 & 13-II-1992     

The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions

We prove existence of Schoen's and other triply periodic minimal surfaces via conjugate (polygonal) Plateau problems. The simpler of these minimal surfaces can be deformed into constant mean curvature surfaces by solving analogous Plateau problems in S³. The required contours in S³ are obtained by working with the great circle orbits of Hopf S¹-actions in the same way as with families of parallel lines in ³. Annular Plateau problems give new embedded minimal surfaces in S³. For many of the minimal surfaces in ³ global Weierstraß representations are derived.Dedicated to Wilhelm Klingenberg     

The geometry of minimal surfaces of finite genus II; nonexistence of one limit end examples

We demonstrate that a properly embedded minimal surface in ³ with finite genus cannot have one limit end. Mathematics Subject Classification (1991) 53A10, 49Q05, 53C42     

The geometry of 1-based minimal types

In this paper, we study the geometry of a (nontrivial) 1-based rank-1 complete type. We show that if the (localized, resp.) geometry of the type is modular, then the (localized, resp.) geometry is projective over a division ring. However, unlike the stable case, we construct a locally modular type that is not affine. For the general 1-based case, we prove that even if the geometry of the type itself is not projective over a division ring, it is when we consider a 2-fold or 3-fold of the geometry altogether. In particular, it follows that in any -categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.

     

The structure of singly-periodic minimal surfaces

The research described in this paper was supported by research grant DE-FG02-86ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grants DMS-8611574 and DMS-8802858     

The doubly periodic Costa surfaces

We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the planar end of the Costa surface by symmetry curves. Received: 14 February 2001; in final form: 24 April 2001 / Published online: 29 April 2002     

The geometry of circumscribing polygons of minimal perimeter

LetC be a closed strictly convex curve in the Euclidean plane, and letC(n) denote the class ofn-sided polygons which circumscribeC. Geometric conditions are given for polygonsP C(n) which have minimum perimeter in the classC(n).