On self-intersections of immersed surfaces

A daisy graph is a union of immersed circles in 3-space which intersect only at the triple points. It is shown that a daisy graph can always be realized as the self-intersection set of an immersed closed surface in 3-space and the surface may be chosen to be orientable if and only if the daisy graph has an even number of edges on each immersed circle.

     

On the control and guidance of the motion of an immersed body: Some problems in stochastic control

In this work, the following problem is considered. A rigid body is moving immersed in a fluid of infinite depth. Its goal is to hit another body which floats on the surface of the fluid. Using stochastic control methods, a sequence of subproblems concerning the guidance and control of the immersed rigid body are dealt with.     

Sufficient conditions for constant mean curvature surfaces to be round

We study under what condition a constant mean curvature surface can be round: i) If the boundary of a compact immersed disk type constant mean curvature surface in consists of lines of curvature and has less than 4 vertices with angle , then the surface is spherical; ii) A compact immersed disk type capillary surface with less than 4 vertices in a domain of bounded by spheres or planes is spherical; iii) The mean curvature vector of a compact embedded capillary hypersurface of with smooth boundary in an unbounded polyhedral domain with unbalanced boundary should point inward; iv) If the kth order () mean curvature of a compact immersed constant mean curvature hypersurface of without boundary is constant, then the hypersurface is a sphere. Received: 3 October 2000 / Published online: 1 February 2002     

The asymptotic blow-up of a surface in Euclidean 3-space

Given a smoothly immersed surface in Euclidean (or affine) 3-space, the asymptotic directions define a subset in the Grassmann bundle of unoriented one-dimensional subspaces over the surface. This links the Euler characteristic of the region where the Gauss curvature is nonpositive with the index of singularities in a natural line field defined on this subset. To apply this we need only identify mechanisms which restrict the index of the singularities. In Section 2.1 we show that specific configurations of nonpositive Gauss curvature cannot be realized by an immersed surface and that specific configurations in 2-sphere cannot be realized as Gauss images of surfaces. In Section 2.2 we prove an existence theorem for surfaces which satisfy regularity conditions and a Symplectic Monge Ampere PDE. In general, a PDE of this type will not restrict the indices of the singularities over a solution. However, we show that over a surface of nonzero constant mean curvature the indices are restricted and, hence, that specific configurations of nonpositive Gauss curvature cannot be realized by a constant mean curvature surface.     

Boundaries of Flat Compact Surfaces in 3-Space

This paper deals with the problem `which knots or links in3-space bound flat (immersed) compact surfaces?' In aforthcoming paper by the author, it is proven that any simple closedspace curve can be deformed until it bounds a flat orientable compact(Seifert) surface. The main results of this paper are that there existknots that do not bound any flat compact surfaces. The lower bound oftotal curvature of a knot bounding an orientable nonnegatively curvedcompact surface can, for varying knot types, be arbitrarily much greaterthan the infimum of curvature needed for the knot to have its knot type.The number of 3-singular points (points of zero curvatureor if not then of zero torsion) on the boundary of a flat immersedcompact surface is greater than or equal to twice the absolute value ofthe Euler characteristic of the surface. A set of necessary and, in aweakened sense, sufficient conditions for a knot or link to be what wecall a generic boundary of a flat immersed compact surface withoutplanar regions is given.     

四元空间形式中具有常数Khler角的浸入曲面

对四元Khler流形中的浸入曲面引入了Khler角的概念,同时讨论Khler角是常数的情形.主要结果是:若x:M→N(c)是具有常数Q-截面曲率c的实四维四元空间形式N(c)中具有常数Khler角θ(sinθ≠0)的等距浸入曲面,则必有c=0.     

Lifespan Theorem for simple constrained surface diffusion flows

We consider closed immersed hypersurfaces in R³ and R⁴ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a Lifespan Theorem for these flows, which gives a positive lower bound on the time for which a smooth solution exists, and a small upper bound on the total curvature during this time. The hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature. This turns out to be a deep property of the initial manifold, as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in L² for M² immersed in R³ and additionally on the concentration in L³ for M³ immersed in R⁴. This is stronger than a previous result on a different class of constrained surface diffusion flows, as here we obtain an improved lower bound on maximal time, a better estimate during this period, and eliminate any assumption on the area of the evolving hypersurface.     

The Gauss map of minimal surfaces in Berger spheres

It is proved that a pair of spinors satisfying a Dirac type equation represents surfaces immersed in Berger spheres with prescribed mean curvature. Using this, we prove that the Gauss map of a minimal surface immersed in a Berger sphere is harmonic. Conversely, we exhibit a representation of minimal surfaces in Berger spheres in terms of a given harmonic map. The examples we constructed appear in associated families.     

Extending immersions of curves to properly immersed surfaces

Proper generic immersions of compact one-dimensional manifolds in surfaces are studied. Suppose an immersion γ of a collection of circles is given with an even number of double points in a closed surface G. Then γ extends to various proper immersions of surfaces in three-manifolds that are bounded by G. Some of these extensions do not have triple points. The minimum of the genera of the triple point free surfaces is an invariant of the curve. An algorithm to compute this invariant is given.Necessary and suffecient conditions determine if a given collection δ of immersed arcs in a surface F maps to the double points set of a proper immersion. In case the conditions are satisfied, an immersion of F into a three-manifold that depends on δ is constructed explicitly. In the process, the possible triple points of immersed surfaces in three-manifolds are categorized.The techniques are applied to find examples of curves in surfaces that do not bound immersed disks in any three-manifold.     

Minimal annuli with constant contact angle along the planar boundaries

We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.     

E~3中闭曲面上的一个积分公式及其应用

利用Minkowski公式,得到了等距浸入在E3中的闭曲面上的一个有趣积分公式,应用此公式得到了闭曲面的一个特征.     

Diameter estimates for submanifolds in manifolds with nonnegative curvature

Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On the other hand, for a compact convex surface with boundary smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we can estimate its intrinsic diameter in terms of its mean curvature field integral and the length of its boundary. These results are supplements of previous work of Topping, Wu-Zheng and Paeng.     

Locally convex surfaces immersed in a Killing submersion

We generalize Hadamard-Stoker-Currier Theorems for surfaces immersed in a Killing submersion over a strictly Hadamard surface whose fibers are the trajectories of a unit Killing field. We prove that every complete surface whose principal curvatures are greater than a certain function (depending on the ambient manifold) at each point, must be properly embedded, homeomorphic either to the sphere or to the plane and, in the latter case, we study the behavior of the end.     

单位球中的极小曲面

Let M be a closed surface with positive Gauss curvature minimally immersed in a standard Euclidean unit sphere S~n.In this paper,we choose a local orthonormal frame field on M,under which the shape operators have very convenient form.We also give some applications of this kind of frame field.     

Constructing isothermal curvature line coordinates on surfaces which admit them

Isothermic parameterizations are synonyms of isothermal curvature line parameterizations, for surfaces immersed in Euclidean spaces. We provide a method of constructing isothermic coordinate charts on surfaces which admit them, starting from an arbitrary chart. One of the primary applications of this work consists of numerical algorithms for surface visualization.     

The Spectrum of the Martin-Morales-Nadirashvili Minimal Surfaces Is Discrete

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of ℝ³ is discrete. This gives a positive answer to a question of Yau (Asian J. Math. 4:235–278, 2000).     

Combinatorial volume preserving flows and taut foliations

A -injective closed surface in an orientable 3-manifold with a tangentially smooth, transversely C ⁰ taut foliation can be homotoped to an immersed surface which is either transverse to the foliation except at isolated saddle tangencies or mapped into a leaf. Received: November 11, 1997     

Justification of an integro-functional method for diffraction problems in the presence of a half-space

In the present paper, we consider the discrete sources method as applied to the analysis of a scalar problem of the diffraction of a plane wave on a particle partly immersed in an absorbing half-space. The convergence of the method is proved with the use of a system of surface integral equations. We prove the Fredholm property of the obtained system and its unique solvability.     

The Bernstein problem for complete Lagrangian stationary surfaces

In this paper, we investigate the global geometric behavior of lagrangian stationary surfaces which are lagrangian surfaces whose area is critical with respect to lagrangian variations. We find that if a complete oriented immersed lagrangian surface has quadratic area growth, one end and finite topological type, then it is minimal and hence holomorphic. The key to the proof is the mean curvature estimate of Schoen and Wolfson combined with the observation that a complete immersed surface of quadratic area growth, finite topology and mean curvature has finite total absolute curvature.

     

Envelopes of holomorphy and holomorphic discs

The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a neighbourhood of an arbitrary point of the envelope of holomorphy can be performed by applying the Continuity Principle once. Another corollary concerns representation of certain elements of the fundamental group of the domain by boundaries of analytic discs. A particular case is the following. Given a contact three-manifold with Stein filling, any element of the fundamental group of the contact manifold whose representatives are contractible in the filling can be represented by the boundary of an immersed analytic disc.