Surfaces Of Finite Total Curvature Research Articles

Systolically extremal nonpositively curved surfaces are flat with finitely many singularities

The regularity of systolically extremal surfaces is a notoriously difficult problem already discussed by Gromov in 1983, who proposed an argument toward the existence of [Formula: see text]-extremizers exploiting the theory of [Formula: see text]-regularity developed by White and others by the 1950s. We propose to study the problem of systolically extremal metrics in the context of generalized metrics of nonpositive curvature. A natural approach would be to work in the class of Alexandrov surfaces of finite total curvature, where one can exploit the tools of the completion provided in the context of Radon measures as studied by Reshetnyak and others. However the generalized metrics in this sense still don’t have enough regularity. Instead, we develop a more hands-on approach and show that, for each genus, every systolically extremal nonpositively curved surface is piecewise flat with finitely many conical singularities. This result exploits a decomposition of the surface into flat systolic bands and nonsystolic polygonal regions, as well as the combinatorial/topological estimates of Malestein–Rivin–Theran, Przytycki, Aougab–Biringer–Gaster and Greene on the number of curves meeting at most once, combined with a kite excision move. The move merges pairs of conical singularities on a surface of genus [Formula: see text] and leads to an asymptotic upper bound [Formula: see text] on the number of singularities.

Interpolation and optimal hitting for complete minimal surfaces with finite total curvature

We prove that, given a compact Riemann surface $$\Sigma $$ and disjoint finite sets $$\varnothing \ne E\subset \Sigma $$ and $$\Lambda \subset \Sigma $$ , every map $$\Lambda \rightarrow \mathbb {R}^3$$ extends to a complete conformal minimal immersion $$\Sigma \setminus E\rightarrow \mathbb {R}^3$$ with finite total curvature. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in $$\mathbb {R}^3$$ with finite total curvature. To this respect we provide, for each integer $$r\ge 1$$ , a set $$A\subset \mathbb {R}^3$$ consisting of $$12r+3$$ points in an affine plane such that if A is contained in a complete nonflat orientable immersed minimal surface $$X:M\rightarrow \mathbb {R}^3$$ , then the absolute value of the total curvature of X is greater than $$4\pi r$$ . In order to prove this result we obtain an upper bound for the number of intersections of a complete immersed minimal surface of finite total curvature in $$\mathbb {R}^3$$ with a straight line not contained in it, in terms of the total curvature and the Euler characteristic of the surface.

Minimal surfaces in minimally convex domains

In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb {R}^3$ can be approximated uniformly on compacts in $\mathring M=M\setminus bM$ by proper complete conformal minimal immersions $\mathring M\to D$. We also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in $\mathbb {R}^3$, and we characterize the minimal surface hull of a compact set $K$ in $\mathbb {R}^n$ for any $n\ge 3$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense.

Minimal surfaces with two ends which have the least total absolute curvature

In this paper, we consider complete non-catenoidal minimal surfaces of finite total curvature with two ends. A family of such minimal surfaces with least total absolute curvature is given. Moreover, we obtain a uniqueness theorem for this family from its symmetries.

Uniform approximation by complete minimal surfaces of finite total curvature in ℝ³

We prove that any compact minimal surface in R 3 \mathbb {R}^3 can be uniformly approximated by complete minimal surfaces of finite total curvature in R 3 \mathbb {R}^3 . This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on regions of finite conformal type. We deal only with the orientable case.

Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$

We consider minimal surfaces $M$ which are complete, embedded, and have finite total curvature in $\mathbb{R}^3$, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation $\Delta u+ f(u) = 0$ in $\mathbb{R}^3$. Here $f = −W'$ with $W$ bi-stable and balanced, for instance $W(u) = \frac{1}{4} (1 − u^2)^2$. We assume that $M$ has $m \ge 2$ ends, and additionally that $M$ is non-degenerate, in the sense that its bounded Jacobi fields are all originated from rigid motions (this is known for instance for a Catenoid and for the Costa-Hoffman-Meeks surface of any genus). We prove that for any small $\alpha \gt 0$, the Allen-Cahn equation has a family of bounded solutions depending on $m − 1$ parameters distinct from rigid motions, whose level sets are embedded surfaces lying close to the blown-up surface $M_\alpha := \alpha^{−1}M$, with ends possibly diverging logarithmically from $M_\alpha$. We prove that these solutions are $L^\infty$-non-degenerate up to rigid motions, and find that their Morse index coincides with the index of the minimal surface. Our construction suggests parallels of De Giorgi conjecture for general bounded solutions of finite Morse index.

Complete embedded minimal surfaces of finite total curvature with planar ends of smallest possible order

Using a method by Traizet (J Differ Geom 60:103–153, 2002), which reduces the construction of minimal surfaces via the Weierstras Theorem and the implicit function theorem to solving algebraic equations in several complex variables, we will show the existence of complete embedded minimal surfaces of finite total curvature with planar ends of least possible order.

Comparison between second variation of area and second variation of energy of a minimal surface

We show that the difference between the Morse index of a closed minimal surface as a critical point of the area functional and its Morse index as a critical point of the energy is at most the real dimension of Teichmüller space. This enables us to bound the index of a closed minimal surface in an arbitrary Riemannian manifold by the area and genus of the surface, and the dimension and geometry of the ambient manifold. Our method also yields surprisingly good upper bounds on the index of a minimal surface of finite total curvature in Euclidean space of any dimension.

Minimal surfaces of finite total curvature in H×R

We prove that finite total curvature minimal surface of H^2xR are characterize by meromorphic extension of Abresch-Rosenberg quadratic differential and the finite total curvature is a multiple of 2pik with integer k. When the surface is a graph, it is asymptotic to a Scherk's type example.

A balancing condition for weak limits of families of minimal surfaces

We prove a balancing condition for weak limits of families of embedded minimal surfaces of finite total curvature. We use it to prove compactness theorems for certain families of minimal surfaces.

An analogue of the UP-iteration for constant mean curvature one surfaces in hyperbolic 3-space

The UP-iteration is a method for constructing new rational minimal surfaces of genus zero from a given rational minimal surface. Here, an analogue of the UP-iteration is given for constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space H 3 . This yields countably many new complete CMC-1 surfaces of finite total curvature in H 3 , which admit isometric deformations preserving principal curvatures.

Existence of algebraic minimal surfaces for an arbitrary puncture set

We will show that any punctured Riemann surface can be conformally immersed into a Euclidean 3-space as a branched complete minimal surface of finite total curvature called an algebraic minimal surface.

Surfaces de Bryant dans [formula omitted] de type fini

We define the notion of Bryant surfaces of finite type: an annular end of a Bryant surface is said to be of finite type if its hyperbolic Gauss map is of finite growth (in the sense of Nevanlinna), and a Bryant surface is said to be of finite type if it is of finite conformal type and if all its ends are of finite type. We prove that a Bryant surface of finite total curvature is of finite type.

Stable minimal surfaces of finite total curvature

Stable minimal surfaces of finite total curvature

Immersions of Finite Geometric Type in Euclidean Spaces

In this paper, we introduce the class of hypersurfaces of finitegeometric type. They are defined as the ones that share the basicdifferential topological properties of minimal surfaces of finite totalcurvature. We extend to surfaces in this class the classical theorem ofOsserman on the number of omitted points of the Gauss mapping ofcomplete minimal surfaces of finite total curvature. We give aclassification of the even-dimensional catenoids as the only even-dimensional minimal hypersurfaces of R n of finite geometric type.

An Embedded Minimal Surface with no Symmetries

We construct embedded minimal surfaces of finite total curvature in euclidean space by gluing catenoids and planes. We use Weierstrass Representation and we solve the Period Problem using the Implicit Function Theorem. As a corollary, we obtain the existence of minimal surfaces with no symmetries.

Constant mean curvature surfaces with Delaunay ends

In this paper we shall present a construction of complete surfaces M in R3 with finitely many ends and finite topology, and with nonzero constant mean curvature (CMC). This construction is parallel to the well-known original construction by Kapouleas [5], but we feel that ours is somewhat simpler analytically, and controls the resulting geometry more closely. On the other hand, the surfaces we construct have a rather different, and usually simpler, geometry than those of Kapouleas; in particular, all of the surfaces constructed here are noncompact, so we do not obtain any of his immersed compact examples. The method we use here closely parallels the one we developed recently [10] to study the very closely related problem of constructing Yamabe metrics on the sphere with k isolated singular points, just as Kapouleas’ construction parallels the earlier construction of singular Yamabe metrics by Schoen [18]. The original examples of noncompact CMC surfaces were those in the one-parameter family of rotationally invariant surfaces discovered by Delaunay in 1841 [2]. One extreme element of this family is the cylinder; the ‘Delaunay surfaces’ are periodic, and the embedded members of this family (which are called unduloids) interpolate between the cylinder and an infinite string of spheres arranged along a common axis. The family continues beyond this, but the elements now are immersed (and are called nodoids). The role of Delaunay surfaces in the theory of complete CMC surfaces is analogous to the role of catenoids (and planes) in the study of complete minimal surfaces of finite total curvature. For example, just as any complete minimal surface with two ends must be a catenoid [19], it was proved by Meeks [14] and Korevaar, Kusner and Solomon [8] that any Alexandrov embedded constant mean curvature surface with at most two ends is necessarily a Delaunay surface. A rather more remarkable theorem, paralleling the fact that any end of a complete minimal surface of finite total curvature must be asymptotic to a catenoid or a plane, is the fact that any embedded end of a CMC surface must be asymptotic to one of these rotationally symmetric Delaunay surfaces (and in particular, must be cylindrically bounded).

A connected sum construction for complete minimal surfaces of finite total curvature

A connected sum construction for complete minimal surfaces of finite total curvature

Compactification of a class of conformally flat 4-manifold

In this paper we generalize Huber’s result on complete surfaces of finite total curvature. For complete locally conformally flat 4-manifolds of positive scalar curvature with Q curvature integrable, where Q is a variant of the Chern-Gauss-Bonnet integrand; we first derive the Cohn-Vossen inequality. We then establish finiteness of the topology. This allows us to provide conformal compactification of such manifolds.

Nonexistence of certain complete minimal surfaces with planar ends

In this paper we prove that two lines bounding an immersed minimal surface in a slab in R3 homeomorphic to a compact Riemann surface minus two disks and a finite number of points must be parallel. This theorem is extended to a higher dimensional minimal hypersurface. Also it is proved that if the Gauss map of a complete embedded minimal surface of finite total curvature at a planar end has order two, then the intersection of the surface with the plane asymptotic to the planar end cannot admit a one-to-one orthogonal projection onto any line in the plane.