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Paul Yang - One of the best experts on this subject based on the ideXlab platform.
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Paneitz operator for metrics near \(S^{3}\)
Calculus of Variations and Partial Differential Equations, 2017Co-Authors: Fengbo Hang, Paul YangAbstract:We derive the first and Second Variation Formula for the Green’s function pole’s value of Paneitz operator on the standard three sphere. In particular it is shown that the first Variation vanishes and the Second Variation is nonpositively definite. Moreover, the Second Variation vanishes only at the direction of conformal deformation. We also introduce a new invariant of the Paneitz operator and illustrate its close relation with the Second eigenvalue and Sobolev inequality of Paneitz operator.
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A Positive Mass Theorem in Three Dimensional Cauchy–Riemann Geometry
Advances in Mathematics, 2017Co-Authors: Jih-hsin Cheng, Andrea Malchiodi, Paul YangAbstract:Abstract We define an ADM-like mass, called p-mass, for an asymptotically flat pseudohermitian manifold. The p-mass for the blow-up of a compact pseudohermitian manifold (with no boundary) is identified with the first nontrivial coefficient in the expansion of the Green function for the CR Laplacian. We deduce an integral Formula for the p-mass, and we reduce its positivity to a solution of Kohn's equation. We prove that the p-mass is non-negative for (blow-ups of) compact 3-manifolds of positive CR Yamabe invariant and with non-negative CR Paneitz operator. Under these assumptions, we also characterize the zero mass case as the standard three dimensional CR sphere. We then show the existence of (non-embeddable) CR 3-manifolds having nonpositive Paneitz operator or negative p-mass through a Second Variation Formula. Finally, we apply our main result to find solutions of the CR Yamabe problem with minimal energy.
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Paneitz operator for metrics near $S^3$
arXiv: Differential Geometry, 2015Co-Authors: Fengbo Hang, Paul YangAbstract:We derive the first and Second Variation Formula for the Green's function pole's value of Paneitz operator on the standard three sphere. In particular it is shown that the first Variation vanishes and the Second Variation is nonpositively definite. Moreover, the Second Variation vanishes only at the direction of conformal deformation. We also introduce a new invariant of the Paneitz operator and illustrate its close relation with the Second eigenvalue and Sobolev inequality of Paneitz operator.
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A positive mass theorem in three dimensional Cauchy-Riemann geometry
arXiv: Differential Geometry, 2013Co-Authors: Jih-hsin Cheng, Andrea Malchiodi, Paul YangAbstract:We define an ADM-like mass, called p-mass, for an asymptotically flat pseudohermitian manifold. The p-mass for the blow-up of a compact pseudohermitian manifold (with no boundary) is identified with the first nontrivial coefficient in the expansion of the Green function for the CR Laplacian. We deduce an integral Formula for the p-mass, and we reduce its positivity to a solution of Kohn's equation. We prove that the p-mass is non-negative for (blow-ups of) compact 3-manifolds of positive Tanaka-Webster class and with non-negative CR Paneitz operator. Under these assumptions, we also characterize the zero mass case as the standard three dimensional CR sphere. We then show the existence of (non-embeddable) CR 3-manifolds having nonpositive Paneitz operator or negative p-mass through a Second Variation Formula. Finally, we apply our main result to find solutions of the CR Yamabe problem with minimal energy.
Tudor S. Ratiu - One of the best experts on this subject based on the ideXlab platform.
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Euler–Poincaré Reduction on Principal Bundles
Letters in Mathematical Physics, 2001Co-Authors: M. Castrillón López, P. L. García Pérez, Tudor S. RatiuAbstract:Let π: P → M be an arbitrary principal G -bundle. We give a full proof of the Euler–Poincaré reduction for a G -invariant Lagrangian L : J ^1 P → R as well as the study of the Second Variation Formula, the conservations laws, and study some of their properties.
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Euler-Poincaré reduction on principal bundles
Letters in Mathematical Physics, 2001Co-Authors: Marco Castrillón López, P. L. García Pérez, Tudor S. RatiuAbstract:Let π: P → M be an arbitrary principal G-bundle. We give a full proof of the Euler–Poincare reduction for a G-invariant Lagrangian L: J1P → R as well as the study of the Second Variation Formula, the conservations laws, and study some of their properties.
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Euler^PoincareReduction on Principal Bundles
2001Co-Authors: P. L. Garciap Erez, Tudor S. RatiuAbstract:Let p: P ! M be an arbitrary principal G-bundle. We give a full proof of the Euler ^ Poincarereduction for a G-invariant Lagrangian L: J 1 P ! R as well as the study of the Second Variation Formula, the conservations laws, and study some of their properties. Mathematics Subject Classi¢cations (2000). 58E30, 53C05.
Nefton Pali - One of the best experts on this subject based on the ideXlab platform.
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The total Second Variation of Perelman’s $$\mathcal W $$ W
Calculus of Variations and Partial Differential Equations, 2014Co-Authors: Nefton PaliAbstract:We show a very simple and general total Second Variation Formula for Perelman’s $$\mathcal W $$ W -functional at arbitrary points in the space of Riemannian metrics. Moreover we perform a study of the properties of the Variations of Kähler structures. We deduce a quite simple and general total Second Variation Formula for Perelman’s $$\mathcal W $$ W -functional with respect to such Variations. In this case the differential term in the Formula depends strongly on the Variation of the complex structure. We discover also convexity of Perelman’s $$\mathcal W $$ W -functional along particular Variations over points with non-negative Bakry-Emery-Ricci tensor.
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A Second Variation Formula for Perelman’s $$\mathcal W $$ W -functional al
Mathematische Zeitschrift, 2014Co-Authors: Nefton PaliAbstract:We show a quite simple Second Variation Formula for Perelman’s $$\mathcal W $$ W -functional along the modified Kähler-Ricci flow over Fano manifolds.
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A Second Variation Formula for Perelman’s \mathcal W -functional along the modified Kähler-Ricci flow
Mathematische Zeitschrift, 2013Co-Authors: Nefton PaliAbstract:We show a quite simple Second Variation Formula for Perelman’s \(\mathcal W \)-functional along the modified Kahler-Ricci flow over Fano manifolds.
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The total Second Variation of Perelman’s \(\mathcal W \)-functional
Calculus of Variations and Partial Differential Equations, 2013Co-Authors: Nefton PaliAbstract:We show a very simple and general total Second Variation Formula for Perelman’s \(\mathcal W \)-functional at arbitrary points in the space of Riemannian metrics. Moreover we perform a study of the properties of the Variations of Kahler structures. We deduce a quite simple and general total Second Variation Formula for Perelman’s \(\mathcal W \)-functional with respect to such Variations. In this case the differential term in the Formula depends strongly on the Variation of the complex structure. We discover also convexity of Perelman’s \(\mathcal W \)-functional along particular Variations over points with non-negative Bakry-Emery-Ricci tensor.
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A Second Variation Formula for Perelman's $\mathcal{W}$-functional along the modified K\"ahler-Ricci flow
arXiv: Differential Geometry, 2012Co-Authors: Nefton PaliAbstract:We show a quite simple Second Variation Formula for Perelman's $\mathcal{W}$-functional along the modified K\"ahler-Ricci flow over Fano manifolds.
Louis Omenyi - One of the best experts on this subject based on the ideXlab platform.
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Conformal Variations of the spectral zeta function of the Laplacian
Journal of Mathematical and Computational Science, 2016Co-Authors: Louis OmenyiAbstract:This work raises and addresses a question about the behaviour of the Variations of the spectral zeta function, ζ g (s), of the Laplacian, ∆ g , on a closed connected smooth Riemannian manifold, (M,g), at any point s = s 0 . We introduce a certain distributional integral kernel and compute a Second Variation Formula of ζ g (s) on closed homogeneous Riemannian manifolds under volume-preserving conformal metric perturbations in terms of the kernel. Some criticality conditions for the spectral Variations are found.
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On the Second Variation of the spectral zeta function of the Laplacian on homogeneous Riemanniann manifolds
2014Co-Authors: Louis OmenyiAbstract:The spectral zeta function, introduced by Minakshisundaram and Pleijel in [36] and denoted by ζg (s), encodes important spectral information for the Laplacian on Riemannian manifolds. For instance, the important notions of the determinant of the Laplacian and Casimir energy are defined via the spectral zeta function. On homogeneous manifolds, it is known that the spectral zeta function is critical with respect to conformal metric perturbations, (see e.g Richardson ([47]) and Okikiolu ([41])). In this thesis, we compute a Second Variation Formula of ζg (s) on closed homogeneous Riemannian manifolds under conformal metric perturbations. It is well known that the quadratic form corresponding to this Second Variation is given by a certain pseudodifferential operator that depends meromorphically on s. The symbol of this operator was analysed by Okikiolu in ([42]). We analyse it in more detail on homogeneous spaces, in particular on the spheres Sn. The case n = 3 is treated in great detail. In order to describe the Second Variation we introduce a certain distributional integral kernel, analyse its meromorphic properties and the pole structure. The Casimir energy defined as the finite part of ζg (− 1/2 ) on the n-sphere and other points of ζg (s) are used to illustrate our results. The techniques employed are heat kernel asymptotics on Riemannian manifolds, the associated meromorphic continuation of the zeta function, harmonic analysis on spheres, and asymptotic analysis.
M. Castrillón López - One of the best experts on this subject based on the ideXlab platform.
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Euler–Poincaré Reduction on Principal Bundles
Letters in Mathematical Physics, 2001Co-Authors: M. Castrillón López, P. L. García Pérez, Tudor S. RatiuAbstract:Let π: P → M be an arbitrary principal G -bundle. We give a full proof of the Euler–Poincaré reduction for a G -invariant Lagrangian L : J ^1 P → R as well as the study of the Second Variation Formula, the conservations laws, and study some of their properties.