scholarly journals The Hele-Shaw flow and moduli of holomorphic discs

2015 ◽  
Vol 151 (12) ◽  
pp. 2301-2328 ◽  
Author(s):  
Julius Ross ◽  
David Witt Nyström
Keyword(s):  
Single Point ◽  
Jordan Domain ◽  
Local Existence ◽  
Laplacian Growth ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Short Time ◽  
Totally Real ◽  
Holomorphic Discs ◽  
Time Existence

We present a new connection between the Hele-Shaw flow, also known as two-dimensional Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this, we prove short-time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also when starting from a smooth Jordan domain. Applying the same ideas, we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.

scholarly journals Remarks on a totally real submanifold

1975 ◽  
Vol 51 (1) ◽  
pp. 5-6 ◽  
Author(s):  
Seiichi Yamaguchi ◽  
Toshihiko Ikawa
Keyword(s):  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Totally Real

Smooth Boundary Values Along Totally Real Submanifolds

1984 ◽  
Vol 36 (2) ◽  
pp. 240-248 ◽  
Author(s):  
Edgar Lee Stout
Keyword(s):  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Regularity Result ◽  
Boundary Values ◽  
Real Submanifolds ◽  
Strongly Pseudoconvex Domain ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Class C ◽  
Totally Real

The main result of this paper is the following regularity result:THEOREM. Let D ⊂ CNbe a bounded, strongly pseudoconvex domain with bD of class Ck, k ≧ 3. Let Σ ⊂ bD be an N-dimensional totally real submanifold, and let f ∊ A(D) satisfy |f| = 1 on Σ, |f| < 1 on. If Σ is of class Cr, 3 ≦ r < k, then the restriction fΣ = f|Σ of f to Σ is of class Cr − 0, and if Σ is of class Ck, then fΣ is of class Ck − 1.Here, of course, A(D) denotes the usual space of functions continuous on , holomorphic on D, and we shall denote by Ak(D), k = 1, 2, …, the space of functions holomorphic on D whose derivatives or order k lie in A(D).

scholarly journals Chen's problem on mixed foliate CR-submanifolds

1989 ◽  
Vol 40 (1) ◽  
pp. 157-160 ◽  
Author(s):  
Mohammed Ali Bashir
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Complex Space Form ◽  
Complex Submanifold ◽  
Hyperbolic Complex ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Simply Connected ◽  
Totally Real ◽  
Cr Submanifolds

We prove that the simply connected compact mixed foliate CR-submanifold in a hyperbolic complex space form is either a complex submanifold or a totally real submanifold. This is the problem posed by Chen.

scholarly journals Totally real pseudo-umbilical submanifolds of a quaternion space form

1998 ◽  
Vol 40 (1) ◽  
pp. 109-115 ◽  
Author(s):  
Huafei Sun
Keyword(s):  
Riemannian Manifold ◽  
Projective Space ◽  
Sectional Curvature ◽  
Space Form ◽  
Real Plane ◽  
Quaternion Space ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Totally Real

Let M(c) denote a 4n-dimensional quaternion space form of quaternion sectional curvature c, and let P(H) denote the 4n-dimensional quaternion projective space of constant quaternion sectional curvature 4. Let N be an n-dimensional Riemannian manifold isometrically immersed in M(c). We call N a totally real submanifold of M(c) if each tangent 2-plane of N is mapped into a totally real plane in M (c). B. Y. Chen and C. S. Houh proved in [1].

On Totally Real Submanifolds in a 6-Sphere

1990 ◽  
Vol 33 (2) ◽  
pp. 162-166
Author(s):  
M. A. Bashir
Keyword(s):  
Complex Structure ◽  
Dimensional Sphere ◽  
Almost Complex Structure ◽  
Real Submanifolds ◽  
Cayley Algebra ◽  
Almost Complex ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Totally Real ◽  
Kaehlerian Manifold

AbstractThe 6-dimensional sphere S6 has an almost complex structure induced by properties of Cayley algebra. With respect to this structure S6 is a nearly Kaehlerian manifold. We investigate 2-dimensional totally real submanifolds in S6. We prove that a 2-dimensional totally real submanifold in S6 is flat.

scholarly journals TOTAL REALITY OF CONORMAL BUNDLES OF HYPERSURFACES IN ALMOST COMPLEX MANIFOLDS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1255-1262 ◽  
Author(s):  
ANDREA SPIRO
Keyword(s):  
Complex Structure ◽  
Almost Complex Structure ◽  
Almost Complex Manifolds ◽  
Almost Complex Manifold ◽  
Conormal Bundle ◽  
Almost Complex ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Totally Real ◽  
Pseudoconvex Hypersurface

A generalization to the almost complex setting of a well-known result by Webster is given. Namely, we prove that if Γ is a strongly pseudoconvex hypersurface in an almost complex manifold (M, J), then the conormal bundle of Γ is a totally real submanifold of (T* M, 𝕁), where 𝕁 is the lifted almost complex structure on T* M defined by Ishihara and Yano.

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