Results 1  10
of
54
An equation of MongeAmpère type in conformal geometry, and fourmanifolds of positive Ricci curvature
, 2004
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Sharp Inequalities, The Functional Determinant, And The Complementary Series
 TRANS. AMER. MATH. SOC
, 1995
"... Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We conce ..."
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Cited by 88 (8 self)
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Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We concentrate especially on the conformally flat case, and obtain formulas in dimensions two, four, and six for the functional determinants of operators which are well behaved under conformal change of metric. The two dimensional formulas are due to Polyakov, and the four dimensional formulas to Branson and rsted; the method is sufficiently streamlined here that we are able to present the six dimensional case for the first time. In particular, we solve the extremal problems for the functional determinants of the conformal Laplacian and of the square of the Dirac operator on S 2 , and in the standard conformal classes on S 4 and S 6 . The S 2 results are due to Onofri, and the S 4 results...
Existence of conformal metrics with constant Qcurvature
, 2004
"... Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Qcurvature under generic assumptions. The problem amounts to solving a fourthorder nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbou ..."
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Cited by 67 (4 self)
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Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Qcurvature under generic assumptions. The problem amounts to solving a fourthorder nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with the compactness result of [31].
Estimates And Extremals For Zeta Function Determinants On FourManifolds
 Commun. Math. Phys
, 1992
"... . Let A be a positive integral power of a natural, conformally covariant differential operator on tensorspinors in a Riemannian manifold. Suppose that A is formally selfadjoint and has positive definite leading symbol. For example, A could be the conformal Laplacian (Yamabe operator) L, or the squ ..."
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Cited by 32 (13 self)
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. Let A be a positive integral power of a natural, conformally covariant differential operator on tensorspinors in a Riemannian manifold. Suppose that A is formally selfadjoint and has positive definite leading symbol. For example, A could be the conformal Laplacian (Yamabe operator) L, or the square of the Dirac operator r= . Within the conformal class fg = e 2w g 0 j w 2 C 1 (M)g of an Einstein, locally symmetric "background" metric g 0 on a compact fourmanifold M , we use an exponential Sobolev inequality of Adams to show that bounds on the functional determinant of A and the volume of g imply bounds on the W 2;2 norm of the conformal factor w, provided that a certain conformally invariant geometric constant k = k(M; g 0 A) is strictly less than 32ß 2 . We show for the operators L and r= 2 that indeed k ! 32ß 2 except when (M; g 0 ) is the standard sphere or a hyperbolic space form. On the sphere, a centering argument allows us to obtain a bound of the same type, de...
Compactness of solutions to some geometric fourthorder equations
, 2005
"... We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minim ..."
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Cited by 29 (8 self)
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We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minimax arguments) of some geometric functional, which give rise to conformal metrics of constant Qcurvature. As a byproduct of our method, we also obtain compactness of such metrics.
Morse theory and a scalar field equation on compact surfaces
"... The aim of this paper is to study a nonlinear scalar field equation on a surface Σ via a Morsetheoretical approach, based on some of the methods in [25]. Employing these ingredients, we derive an alternative and direct proof (plus a clear interpretation) of a degree formula obtained in [18], whic ..."
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Cited by 25 (7 self)
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The aim of this paper is to study a nonlinear scalar field equation on a surface Σ via a Morsetheoretical approach, based on some of the methods in [25]. Employing these ingredients, we derive an alternative and direct proof (plus a clear interpretation) of a degree formula obtained in [18], which used refined blowup estimates from [34] and [17]. Related results are derived for the prescribed Qcurvature equation on four manifolds.
MODULAR CURVATURE FOR NONCOMMUTATIVE TWOTORI
, 1110
"... Abstract. Starting from the description of the conformal geometry of noncommutative 2tori in the framework of modular spectral triples, we explicitly compute the local curvature functionals determined by the value at zero of the zeta functions affiliated with these spectral triples. We give a close ..."
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Cited by 21 (0 self)
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Abstract. Starting from the description of the conformal geometry of noncommutative 2tori in the framework of modular spectral triples, we explicitly compute the local curvature functionals determined by the value at zero of the zeta functions affiliated with these spectral triples. We give a closed formula for the RaySinger analytic torsion in terms of the Dirichlet quadratic form and the generating function for Bernoulli numbers applied to the modular operator. The gradient of the RaySinger analytic torsion is then expressed in terms of these functionals, and yields the analogue of scalar curvature. Computing this gradient in two ways elucidates the meaning of the complicated two variable functions occurring in the formula for the scalar curvature. Moreover, the corresponding evolution equation for the metric produces the appropriate analogue of Ricci curvature. We prove the analogue of the classical result which asserts that in every conformal class the maximum value of the determinant of the Laplacian on metrics of a fixed area is attained only at the constant curvature metric.
Conformal invariants and partial differential equations
 Bull. Amer. Math. Soc. (N.S
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