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Doubly connected minimal surfaces and extremal harmonic mappings
 J. Geom. Anal
"... Abstract. The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial confor ..."
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Abstract. The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Grötzsch and Johannes C. C. Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Björling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial
The space of complete minimal surfaces with finite total curvature as Lagrangian submanifold
 Trans. Amer. Math. Soc
"... Abstract. The space M of nondegenerate, properly embedded minimal surfaces in R 3 with finite total curvature and fixed topology is an analytic lagrangian submanifold of C n,wherenis the number of ends of the surface. In this paper we give two expressions, one integral and the other pointwise, for t ..."
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Abstract. The space M of nondegenerate, properly embedded minimal surfaces in R 3 with finite total curvature and fixed topology is an analytic lagrangian submanifold of C n,wherenis the number of ends of the surface. In this paper we give two expressions, one integral and the other pointwise, for the second fundamental form of this submanifold. We also consider the compact boundary case, and we show that the space of stable nonflat minimal annuli that bound a fixed convex curve in a horizontal plane, having a horizontal end of finite total curvature, is a locally convex curve in the plane C. 1.
The Space of Minimal Annuli Bounded by an Extremal Pair of Planar Curves
 Comm. Anal. Geom
, 1993
"... this paper was supported by research grant DEFG0286ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grant DMS8900285. y ..."
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Cited by 4 (2 self)
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this paper was supported by research grant DEFG0286ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grant DMS8900285. y
Classical open problems in differential geometry
, 2004
"... By a classical problem in differential geometry I mean one which involves smooth curves or surfaces in three dimensional Euclidean space. We list here a number of such problems. For other problems in differential geometry or geometric analysis see [40]. Some problems and many references may also be ..."
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By a classical problem in differential geometry I mean one which involves smooth curves or surfaces in three dimensional Euclidean space. We list here a number of such problems. For other problems in differential geometry or geometric analysis see [40]. Some problems and many references may also be found in [6]. A large collection of problems in discrete and convex geometry may be found in [9]. Also see [13] for nice problems invloving convex bodies. For some problems in geometric knot theory see [2].
unknown title
, 1997
"... The space of complete minimal surfaces with finite total curvature as lagrangian submanifold ..."
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The space of complete minimal surfaces with finite total curvature as lagrangian submanifold
Properly embedded minimal annuli bounded by a convex curve
, 2000
"... Abstract.We prove that given a convex Jordan curve Γ ⊂ {x3 = 0}, the space of properly embedded minimal annuli in the halfspace {x3 ≥ 0}, with boundary Γ is diffeomorphic to the interval [0,∞). Moreover, for a fixed positive number a, the exterior Plateau problem that consists of finding a properly ..."
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Abstract.We prove that given a convex Jordan curve Γ ⊂ {x3 = 0}, the space of properly embedded minimal annuli in the halfspace {x3 ≥ 0}, with boundary Γ is diffeomorphic to the interval [0,∞). Moreover, for a fixed positive number a, the exterior Plateau problem that consists of finding a properly embedded minimal annulus in the upper halfspace, with finite total curvature, boundary Γ and a catenoid type end with logarithmic growth a has exactly zero, one or two solutions, each one with a different stability character for the Jacobi operator.
ON THE TOTAL CURVATURE OF MINIMAL ANNULI IN R 3 AND NITSCHE’S CONJECTURE
, 1997
"... Abstract. In this paper we prove the generalized Nitsche’s conjecture proposed by W. H. Meeks III and H. Rosenberg: For t ≥ 0, let Pt denote the horizontal plane of height t over the x1, x2plane. Suppose that M ⊂ R 3 is a minimal annulus with ∂M ⊂ P0 and that M intersects every Pt in a simple close ..."
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Abstract. In this paper we prove the generalized Nitsche’s conjecture proposed by W. H. Meeks III and H. Rosenberg: For t ≥ 0, let Pt denote the horizontal plane of height t over the x1, x2plane. Suppose that M ⊂ R 3 is a minimal annulus with ∂M ⊂ P0 and that M intersects every Pt in a simple closed curve. Then M has finite total curvature. As a consequence, we show that every properly embedded minimal surface of finite topology in R 3 with more than one end has finite total curvature. 1. Introduction. The geometry and topology of minimal annuli in R 3 has attracted geometer’s attention for a long time. For example, B. Riemann described all minimal annuli in R3 that are foliated by the parallel circles, in terms of elliptic functions (see[9]); M. Shiffman [10] studied minimal surfaces in R3 bounded by two parallel convex curves. In 1962, Nitsche
On minimal surfaces bounded by two convex curves in parallel planes
, 2008
"... Abstract. We prove that a compact minimal surface bounded by two closed convex curves in parallel planes close enough to each other must be topologically an annulus. 1 ..."
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Abstract. We prove that a compact minimal surface bounded by two closed convex curves in parallel planes close enough to each other must be topologically an annulus. 1
On the Gauss map of embedded minimal tubes 1
, 903
"... Abstract. A surface is called a tube if its levelsets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the socalled flow vector. We study how the geometry of the Gaussian image of a higherdimensional minimal tube M i ..."
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Abstract. A surface is called a tube if its levelsets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the socalled flow vector. We study how the geometry of the Gaussian image of a higherdimensional minimal tube M is controlled by the angle α(M) between the axis and the flow vector of M. We prove that the diameter of the Gauss image of M is at least 2α(M) if the angle α(M) is positive. As a consequence we derive an estimate on the length of a twodimensional minimal tube M in terms of α(M) and the total Gaussian curvature of M.