The Experts below are selected from a list of 228 Experts worldwide ranked by ideXlab platform
Dehua Wang - One of the best experts on this subject based on the ideXlab platform.
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Isometric Immersion of complete surfaces with slowly decaying negative gauss curvature
arXiv: Differential Geometry, 2016Co-Authors: Feimin Huang, Dehua WangAbstract:The Isometric Immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the Isometric Immersion problem is considered in this paper through the Gauss-Codazzi system for the second fundamental forms. It is shown that if the Gauss curvature satisfies an integrability condition, the surface has a global smooth Isometric Immersion in the three-dimensional Euclidean space even if the Gauss curvature decays very slowly at infinity. The new idea of the proof is based on the novel observations on the decay properties of the Riemann invariants of the Gauss-Codazzi system. The weighted Riemann invariants are introduced and a comparison principle is applied with properly chosen control functions.
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Isometric Immersion of surface with negative gauss curvature and the lax friedrichs scheme
arXiv: Analysis of PDEs, 2015Co-Authors: Feimin Huang, Dehua WangAbstract:The Isometric Immersion of two-dimensional Riemannian manifold with negative Gauss curvature into the three-dimensional Euclidean space is considered through the Gauss-Codazzi equations for the first and second fundamental forms. The large $L^\infty$ solution is obtained which leads to a $C^{1,1}$ Isometric Immersion. The approximate solutions are constructed by the the Lax-Friedrichs finite-difference scheme with the fractional step. The uniform estimate is established by studying the equations satisfied by the Riemann invariants and using the sign of the nonlinear part. The $H^{-1}$ compactness is also derived. A compensated compactness framework is applied to obtain the existence of large $L^\infty$ solution to the Gauss-Codazzi equations for the surfaces more general than those in literature.
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Isometric Immersions of Surfaces with Two Classes of Metrics and Negative Gauss Curvature
Archive for Rational Mechanics and Analysis, 2015Co-Authors: Feimin Huang, Dehua WangAbstract:The Isometric Immersion of two-dimensional Riemannian manifolds or surfaces with negative Gauss curvature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with large data in \({L^{\infty}}\) are obtained through the vanishing viscosity method and the compensated compactness framework. The \({L^{\infty}}\) uniform estimate and H−1 compactness are established through a transformation of state variables and construction of proper invariant regions for two types of given metrics including the catenoid type and the helicoid type. The global weak solutions in \({L^{\infty}}\) to the Gauss-Codazzi equations yield the C1,1 Isometric Immersions of surfaces with the given metrics.
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Isometric Immersions and compensated compactness
Communications in Mathematical Physics, 2010Co-Authors: Gui Qiang Chen, Marshall Slemrod, Dehua WangAbstract:A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as Isometric Immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for Isometric Immersions in $\R^3$. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the Isometric realization of two-dimensional surfaces in $\R^3$. As a first application of this approach, we study the Isometric Immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a $C^{1,1}$ Isometric Immersion of the two-dimensional manifold in $\R^3$ satisfying our prescribed initial conditions. T
Keti Tenenblat - One of the best experts on this subject based on the ideXlab platform.
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Local Isometric Immersions of Pseudo-Spherical Surfaces and Evolution Equations
Hamiltonian Partial Differential Equations and Applications, 2020Co-Authors: Nabil Kahouadji, Niky Kamran, Keti TenenblatAbstract:The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat (Stud. Appl. Math. 74, 55–83, 1986), is characterized by the property that to each solution of a differential equation within the class, there corresponds a two-dimensional Riemannian metric of curvature equal to − 1. The class of differential equations describing pseudo-spherical surfaces carries close ties to the property of complete integrability, as manifested by the existence of infinite hierarchies of conservation laws and associated linear problems. As such, it contains many important known examples of integrable equations, like the sine-Gordon, Liouville and KdV equations. It also gives rise to many new families of integrable equations. The question we address in this paper concerns the local Isometric Immersion of pseudo-spherical surfaces in E3 from the perspective of the differential equations that give rise to the metrics. Indeed, a classical theorem in the differential geometry of surfaces states that any pseudo-spherical surface can be locally Isometrically immersed in E3. In the case of the sine-Gordon equation, one can derive an expression for the second fundamental form of the Immersion that depends only on a jet of finite order of the solution of the pde. A natural question is to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudo-spherical surfaces. In an earlier paper (Kahouadji et al., Second-order equations and local Isometric Immersions of pseudo-spherical surfaces, 25 pp. [arXiv:1308.6545], to appear in Comm. Analysis and Geometry (2015), we have shown that this property fails to hold for all other second order equations, except for those belonging to a very special class of evolution equations. In the present paper, we consider a class of evolution equations for u(x, t) of order k ≥ 3 describing pseudo-spherical surfaces. We show that whenever an Isometric Immersion in E3 exists, depending on a jet of finite order of u, then the coefficients of the second fundamental form are universal, that is they are functions of the independent variables x and t only.
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Local Isometric Immersions of pseudo-spherical surfaces and kth order evolution equations
Communications in Contemporary Mathematics, 2019Co-Authors: Nabil Kahouadji, Niky Kamran, Keti TenenblatAbstract:We consider the class of evolution equations of the form [Formula: see text], [Formula: see text], that describe pseudo-spherical surfaces. These were classified by Chern and Tenenblat in [Pseudospherical surfaces and evolution equations, Stud. Appl. Math 74 (1986) 55–83.]. This class of equations is characterized by the property that to each solution of such an equation, there corresponds a 2-dimensional Riemannian metric of constant curvature [Formula: see text]. Motivated by the special properties of the sine-Gordon equation, we investigate the following problem: given such a metric, is there a local Isometric Immersion in [Formula: see text] such that the coefficients of the second fundamental form of the immersed surface depend on a jet of finite order of [Formula: see text]? We extend our earlier results for second-order evolution equations [N. Kahouadji, N. Kamran and K. Tenenblat, Local Isometric Immersions of pseudo-spherical surfaces and evolution equations, Fields Inst. Commun. 75 (2015) 369–381; N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local Isometric Immersions of pseudo-spherical surfaces, Comm. Anal. Geom. 24(3) (2016) 605–643.] to [Formula: see text]th order equations by proving that there is only one type of equation that admit such an Isometric Immersion. More precisely, we prove under the condition of finite jet dependency that the coefficients of the second fundamental forms of the local Isometric Immersion determined by the solutions [Formula: see text] are universal, i.e. they are independent of [Formula: see text]. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the Isometric Immersion.
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Local Isometric Immersions of pseudo-spherical surfaces and k-th order evolution equations
arXiv: Differential Geometry, 2017Co-Authors: Nabil Kahouadji, Niky Kamran, Keti TenenblatAbstract:We consider the class of evolution equations that describe pseudo-spherical surfaces of the form u\_t = F (u, $\partial$u/$\partial$x, ..., $\partial$^k u/$\partial$x^k), k $\ge$ 2 classified by Chern-Tenenblat. This class of equations is characterized by the property that to each solution of a differential equation within this class, there corresponds a 2-dimensional Riemannian metric of curvature-1. We investigate the following problem: given such a metric, is there a local Isometric Immersion in R 3 such that the coefficients of the second fundamental form of the surface depend on a jet of finite order of u? By extending our previous result for second order evolution equation to k-th order equations, we prove that there is only one type of equations that admit such an Isometric Immersion. We prove that the coefficients of the second fundamental forms of the local Isometric Immersion determined by the solutions u are universal, i.e., they are independent of u. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the Isometric Immersion.
Nabil Kahouadji - One of the best experts on this subject based on the ideXlab platform.
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Local Isometric Immersions of Pseudo-Spherical Surfaces and Evolution Equations
Hamiltonian Partial Differential Equations and Applications, 2020Co-Authors: Nabil Kahouadji, Niky Kamran, Keti TenenblatAbstract:The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat (Stud. Appl. Math. 74, 55–83, 1986), is characterized by the property that to each solution of a differential equation within the class, there corresponds a two-dimensional Riemannian metric of curvature equal to − 1. The class of differential equations describing pseudo-spherical surfaces carries close ties to the property of complete integrability, as manifested by the existence of infinite hierarchies of conservation laws and associated linear problems. As such, it contains many important known examples of integrable equations, like the sine-Gordon, Liouville and KdV equations. It also gives rise to many new families of integrable equations. The question we address in this paper concerns the local Isometric Immersion of pseudo-spherical surfaces in E3 from the perspective of the differential equations that give rise to the metrics. Indeed, a classical theorem in the differential geometry of surfaces states that any pseudo-spherical surface can be locally Isometrically immersed in E3. In the case of the sine-Gordon equation, one can derive an expression for the second fundamental form of the Immersion that depends only on a jet of finite order of the solution of the pde. A natural question is to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudo-spherical surfaces. In an earlier paper (Kahouadji et al., Second-order equations and local Isometric Immersions of pseudo-spherical surfaces, 25 pp. [arXiv:1308.6545], to appear in Comm. Analysis and Geometry (2015), we have shown that this property fails to hold for all other second order equations, except for those belonging to a very special class of evolution equations. In the present paper, we consider a class of evolution equations for u(x, t) of order k ≥ 3 describing pseudo-spherical surfaces. We show that whenever an Isometric Immersion in E3 exists, depending on a jet of finite order of u, then the coefficients of the second fundamental form are universal, that is they are functions of the independent variables x and t only.
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Local Isometric Immersions of pseudo-spherical surfaces and kth order evolution equations
Communications in Contemporary Mathematics, 2019Co-Authors: Nabil Kahouadji, Niky Kamran, Keti TenenblatAbstract:We consider the class of evolution equations of the form [Formula: see text], [Formula: see text], that describe pseudo-spherical surfaces. These were classified by Chern and Tenenblat in [Pseudospherical surfaces and evolution equations, Stud. Appl. Math 74 (1986) 55–83.]. This class of equations is characterized by the property that to each solution of such an equation, there corresponds a 2-dimensional Riemannian metric of constant curvature [Formula: see text]. Motivated by the special properties of the sine-Gordon equation, we investigate the following problem: given such a metric, is there a local Isometric Immersion in [Formula: see text] such that the coefficients of the second fundamental form of the immersed surface depend on a jet of finite order of [Formula: see text]? We extend our earlier results for second-order evolution equations [N. Kahouadji, N. Kamran and K. Tenenblat, Local Isometric Immersions of pseudo-spherical surfaces and evolution equations, Fields Inst. Commun. 75 (2015) 369–381; N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local Isometric Immersions of pseudo-spherical surfaces, Comm. Anal. Geom. 24(3) (2016) 605–643.] to [Formula: see text]th order equations by proving that there is only one type of equation that admit such an Isometric Immersion. More precisely, we prove under the condition of finite jet dependency that the coefficients of the second fundamental forms of the local Isometric Immersion determined by the solutions [Formula: see text] are universal, i.e. they are independent of [Formula: see text]. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the Isometric Immersion.
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Local Isometric Immersions of pseudo-spherical surfaces and k-th order evolution equations
arXiv: Differential Geometry, 2017Co-Authors: Nabil Kahouadji, Niky Kamran, Keti TenenblatAbstract:We consider the class of evolution equations that describe pseudo-spherical surfaces of the form u\_t = F (u, $\partial$u/$\partial$x, ..., $\partial$^k u/$\partial$x^k), k $\ge$ 2 classified by Chern-Tenenblat. This class of equations is characterized by the property that to each solution of a differential equation within this class, there corresponds a 2-dimensional Riemannian metric of curvature-1. We investigate the following problem: given such a metric, is there a local Isometric Immersion in R 3 such that the coefficients of the second fundamental form of the surface depend on a jet of finite order of u? By extending our previous result for second order evolution equation to k-th order equations, we prove that there is only one type of equations that admit such an Isometric Immersion. We prove that the coefficients of the second fundamental forms of the local Isometric Immersion determined by the solutions u are universal, i.e., they are independent of u. Moreover, we show that there exists a foliation of the domain of the parameters of the surface by straight lines with the property that the mean curvature of the surface is constant along the images of these straight lines under the Isometric Immersion.
Hajime Urakawa - One of the best experts on this subject based on the ideXlab platform.
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Harmonic Maps and Bi-Harmonic Maps on CR-Manifolds and Foliated Riemannian Manifolds
Journal of Applied Mathematics and Physics, 2016Co-Authors: Shinji Ohno, Takashi Sakai, Hajime UrakawaAbstract:This is a survey on our recent works on bi-harmonic maps on CR-manifolds and foliated Riemannian manifolds, and also a research paper on bi-harmonic maps principal G-bundles. We will show, (1) for a complete strictly pseudoconvex CR manifold , every pseudo bi-harmonic Isometric Immersion into a Riemannian manifold of non-positive curvature, with finite energy and finite bienergy, must be pseudo harmonic; (2) for a smooth foliated map of a complete, possibly non-compact, foliated Riemannian manifold into another foliated Riemannian manifold, of which transversal sectional curvature is non-positive, we will show that if it is transversally bi-harmonic map with the finite energy and finite bienergy, then it is transversally harmonic; (3) we will claim that the similar result holds for principal G-bundle over a Riemannian manifold of negative Ricci curvature.
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Triharmonic Isometric Immersions into a manifold of non-positively constant curvature
Monatshefte für Mathematik, 2014Co-Authors: Shun Maeta, Nobumitsu Nakauchi, Hajime UrakawaAbstract:A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifolds. We study the generalized Chen’s conjecture for a triharmonic Isometric Immersion \(\varphi \) into a space form of non-positive constant curvature. We show that if the domain is complete and both the 4-energy of \(\varphi \), and the \(L^4\)-norm of the tension field \(\tau (\varphi )\), are finite, then such an Immersion \(\varphi \) is minimal.
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Triharmonic Isometric Immersions into a manifold of non-positively constant curvature
arXiv: Differential Geometry, 2013Co-Authors: Shun Maeta, Nobumitsu Nakauchi, Hajime UrakawaAbstract:A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifolds. We study a triharmonic Isometric Immersion into a space form of non-positively constant curvature. We show that if the domain is complete and both the 4-enegy and the L^4-norm of the tension field are finite, then such an Immersion is minimal.
Alexander Pigazzini - One of the best experts on this subject based on the ideXlab platform.
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ON TOTALLY UMBILICAL Isometric Immersion WITH A SPECIAL KIND OF LIOUVILLE’S METRIC IN THE R^{3} SPACE
arXiv: Differential Geometry, 2018Co-Authors: Alexander PigazziniAbstract:The aim of this short article is to investigate the possibility of existence of totally umbilical Isometric Immersions in R^3 with isothermal parametrization and harmonic metric.
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on totally umbilical Isometric Immersion with a special kind of liouville s metric in the r 3 space
arXiv: Differential Geometry, 2018Co-Authors: Alexander PigazziniAbstract:The aim of this short article is to investigate the possibility of existence of totally umbilical Isometric Immersions in R^3 with isothermal parametrization and harmonic metric.
