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377
The Bernstein problem in the Heisenberg group
, 2005
"... We establish the following theorem of Bernstein type for the first Heisenberg group H¹: Let S be a C² connected Hminimal surface which is a graph over some plane P, then S is either a noncharacteristic vertical plane, or its generalized seed curve satisfies a type of constant curvature condition ..."
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Cited by 42 (10 self)
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We establish the following theorem of Bernstein type for the first Heisenberg group H¹: Let S be a C² connected Hminimal surface which is a graph over some plane P, then S is either a noncharacteristic vertical plane, or its generalized seed curve satisfies a type of constant curvature
The Bernstein problem for intrinsic graphs in the Heisenberg group and calibrations
, 2006
"... In this paper we deal with some problems concerning minimal hypersurfaces in CarnotCarathéodory (CC) structures. More precisely we will introduce a general calibration method in this setting and we will study the Bernstein problem for entire regular intrinsic minimal graphs in a meaningful and simp ..."
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Cited by 25 (3 self)
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In this paper we deal with some problems concerning minimal hypersurfaces in CarnotCarathéodory (CC) structures. More precisely we will introduce a general calibration method in this setting and we will study the Bernstein problem for entire regular intrinsic minimal graphs in a meaningful
Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space
, 2007
"... We classify the entire minimal vertical graphs in the Heisenberg group Nil3 endowed with a Riemannian leftinvariant metric. This classification, which provides a solution to the Bernstein problem in Nil3, is given in terms of the AbreschRosenberg holomorphic differential for minimal surfaces in Ni ..."
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Cited by 15 (3 self)
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We classify the entire minimal vertical graphs in the Heisenberg group Nil3 endowed with a Riemannian leftinvariant metric. This classification, which provides a solution to the Bernstein problem in Nil3, is given in terms of the AbreschRosenberg holomorphic differential for minimal surfaces
A BERNSTEIN PROBLEM IN WARPED PRODUCTS
, 2016
"... Abstract. Uniqueness and nonexistence of entire solutions to the minimal surface equation in warped products R 2 × f R are provided. As a consequence of our results, the classical Bernstein's Theorem is extended. ..."
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Abstract. Uniqueness and nonexistence of entire solutions to the minimal surface equation in warped products R 2 × f R are provided. As a consequence of our results, the classical Bernstein's Theorem is extended.
A solution to the SchroederBernstein problem for Banach spaces
 Bull. London Math. Soc
, 1996
"... Two nonisomorphic Banach spaces are constructed, such that either is a complemented subspace of the other. Moreover, the spaces are of the form Z and Z@Z, where Z is a space isomorphic to Z © Z © Z. The following question arises naturally in the context of Pelczynski's decomposition method. I ..."
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Cited by 16 (2 self)
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to as the SchroederBernstein problem for Banach spaces. If one omits the word 'complemented', one obtains a more basic and natural question, but one that is easily answered with a counterexample. For further details on this and many other aspects of the problem, the reader is referred to a survey
A Bernstein problem for special Lagrangian equation
 Department of Mathematics, Columbia University
"... In this paper we derive a Bernstein type result for the special Lagrangian equation (1.1) F ( D 2 u) = arctan λ1 + · · · + arctan λn = c, ..."
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Cited by 23 (7 self)
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In this paper we derive a Bernstein type result for the special Lagrangian equation (1.1) F ( D 2 u) = arctan λ1 + · · · + arctan λn = c,
Minimal surfaces in pseudohermitian geometry and the Bernstein problem in the Heisenberg group
, 2004
"... We develop a surface theory in pseudohermitian geometry. We define a notion of (p)mean curvature and the associated (p)minimal surfaces. As a differential equation, the pminimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate the go through theo ..."
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Cited by 61 (10 self)
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theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and hence solves the analogue of the Bernstein problem in the Heisenberg group H1. In H1, identified with the Euclidean space R 3, the pminimal surfaces
The regularity of harmonic maps into spheres and applications to Bernstein problems
, 2009
"... ..."
The Bernstein Problem for Complete Lagrangian Stationary Surfaces
"... Abstract. 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 f ..."
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Cited by 1 (0 self)
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Abstract. 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 L 2 mean curvature has finite total absolute curvature. 1.
A NEGATIVE ANSWER TO THE BERNSTEIN PROBLEM FOR INTRINSIC GRAPHS IN THE HEISENBERG GROUP
"... Abstract. A negative answer to the Bernstein problem for entire Hperimeter minimizing intrinsic graphs is given in the setting of the first Heisenberg group H 1 endowed with its CarnotCarathéodory metric structure. Moreover, in all Heisenberg groups H n an area formula for intrinsic graphs with So ..."
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Cited by 6 (0 self)
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Abstract. A negative answer to the Bernstein problem for entire Hperimeter minimizing intrinsic graphs is given in the setting of the first Heisenberg group H 1 endowed with its CarnotCarathéodory metric structure. Moreover, in all Heisenberg groups H n an area formula for intrinsic graphs
Results 1  10
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377