A Monge-Ampère equation in conformal geometry

Matthew J. Gursky

Displaying similar documents to “A Monge-Ampère equation in conformal geometry”

On the Monge-Ampere differential equation $r t - s^{2} + λ^{2} (x, y) = 0$

B. N. Rachajsky (1969)

Matematički Vesnik

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A priori estimates for weak solutions of complex Monge-Ampère equations

Slimane Benelkourchi, Vincent Guedj, Ahmed Zeriahi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $X$ be a compact Kähler manifold and $ω$ be a smooth closed form of bidegree $(1, 1)$ which is nonnegative and big. We study the classes $ℰ_{χ} (X, ω)$ of $ω$ -plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $χ$ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class $ℰ_{χ} (X, ω)$ . This is done by...

A regularity theorem for the complex Monge-Ampère equation in ℂPⁿ

Sławomir Kołodziej (2003)

Annales Polonici Mathematici

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$C^{1, 1}$ regularity of the solutions of the complex Monge-Ampère equation in ℂPⁿ with the n-root of the right hand side in $C^{1, 1}$ is proved.

On a Monge-Ampère type equation in the Cegrell class $_{χ}$

Rafał Czyż (2010)

Annales Polonici Mathematici

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Let Ω be a bounded hyperconvex domain in ℂn and let μ be a positive and finite measure which vanishes on all pluripolar subsets of Ω. We prove that for every continuous and strictly increasing function χ:(-∞,0) → (-∞,0) there exists a negative plurisubharmonic function u which solves the Monge-Ampère type equation $- χ (u) (d d^{c} u) ⁿ = d μ$ . Under some additional assumption the solution u is uniquely determined.

Hölder continuous solutions to Monge–Ampère equations

Jean-Pierre Demailly, Sławomir Dinew, Vincent Guedj, Pham Hoang Hiep, Sławomir Kołodziej, Ahmed Zeriahi (2014)

Journal of the European Mathematical Society

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Let $(X, ω)$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^{p}$ right hand side, $p > 1$ . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $(X, ω)$ of the complex Monge-Ampère operator acting on $ω$ -plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with $L^{p}$ -density belong to $(X, ω)$ and proving that...

Boundary values of functions in Cegrell’s class $_{ψ}$

Pham Hoang Hiep (2007)

Annales Polonici Mathematici

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We study boundary values of functions in Cegrell’s class $_{ψ}$ .

A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$ -curvature

Matthew J. Gursky, Andrea Malchiodi (2015)

Journal of the European Mathematical Society

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In this paper we consider Riemannian manifolds $(M^{n}, g)$ of dimension $n \geq 5$ , with semi-positive $Q$ -curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive $Q$ -curvature. Modifying the test function construction of Esposito-Robert,...

The comparison principle and Dirichlet problem in the class $_{p} (f)$ , p > 0

Pham Hoang Hiep (2006)

Annales Polonici Mathematici

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We establish the comparison principle in the class $_{p} (f)$ . The result obtained is applied to the Dirichlet problem in $_{p} (f)$ .

Conformal Killing graphs in foliated Riemannian spaces with density: rigidity and stability

Marco L. A. Velásquez, André F. A. Ramalho, Henrique F. de Lima, Márcio S. Santos, Arlandson M. S. Oliveira (2021)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we investigate the geometry of conformal Killing graphs in a Riemannian manifold ${\bar{M}}_{f}^{n + 1}$ endowed with a weight function $f$ and having a closed conformal Killing vector field $V$ with conformal factor $ψ_{V}$ , that is, graphs constructed through the flow generated by $V$ and which are defined over an integral leaf of the foliation $V^{⊥}$ orthogonal to $V$ . For such graphs, we establish some rigidity results under appropriate constraints on the $f$ -mean curvature. Afterwards, we obtain some stability...

Concerning the energy class $_{p}$ for 0 < p < 1

Per Åhag, Rafał Czyż, Pham Hoàng Hiêp (2007)

Annales Polonici Mathematici

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The energy class $_{p}$ is studied for 0 < p < 1. A characterization of certain bounded plurisubharmonic functions in terms of $ℱ_{p}$ and its pluricomplex p-energy is proved.

The gradient lemma

Urban Cegrell (2007)

Annales Polonici Mathematici

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We show that if a decreasing sequence of subharmonic functions converges to a function in $W_{l o c}^{1, 2}$ then the convergence is in $W_{l o c}^{1, 2}$ .

Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

Erwann Aubry, Colin Guillarmou (2011)

Journal of the European Mathematical Society

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For odd-dimensional Poincaré–Einstein manifolds $(X^{n + 1}, g)$ , we study the set of harmonic $k$ -forms (for $k < n / 2$ ) which are $C^{m}$ (with $m \in ℕ$ ) on the conformal compactification $\bar{X}$ of $X$ . This set is infinite-dimensional for small $m$ but it becomes finite-dimensional if $m$ is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology $H^{k} (\bar{X}, \partial \bar{X})$ and the kernel of the Branson–Gover [3] differential operators $(L_{k}, G_{k})$ on the conformal infinity $(\partial \bar{X}, [h_{0}])$ . We also relate the set of $C^{n - 2 k + 1} (Λ^{k} (\bar{X}))$ forms in the kernel of $d + δ_{g}$ ...

A Study on $φ$ -recurrence $τ$ -curvature tensor in $(k, μ)$ -contact metric manifolds

Gurupadavva Ingalahalli, C.S. Bagewadi (2018)

Communications in Mathematics

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In this paper we study $φ$ -recurrence $τ$ -curvature tensor in $(k, μ)$ -contact metric manifolds.

Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Zdeněk Dušek (2015)

Czechoslovak Mathematical Journal

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Let $(M, g)$ be a 4-dimensional Einstein Riemannian manifold. At each point $p$ of $M$ , the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor $R$ at $p$ . In this basis, up to standard symmetries and antisymmetries, just $5$ components of the curvature tensor $R$ are nonzero. For the space of constant curvature, the group $O (4)$ acts as a transformation group between ST bases at $T_{p} M$ and for the so-called 2-stein curvature tensors, the group $Sp (1) \subset SO (4)$ acts as a transformation...

Conformal Ricci Soliton in Lorentzian $α$ -Sasakian Manifolds

Tamalika Dutta, Nirabhra Basu, Arindam BHATTACHARYYA (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper we have studied conformal curvature tensor, conharmonic curvature tensor, projective curvature tensor in Lorentzian $α$ -Sasakian manifolds admitting conformal Ricci soliton. We have found that a Weyl conformally semi symmetric Lorentzian $α$ -Sasakian manifold admitting conformal Ricci soliton is $η$ -Einstein manifold. We have also studied conharmonically Ricci symmetric Lorentzian $α$ -Sasakian manifold admitting conformal Ricci soliton. Similarly we have proved that a Lorentzian...

The almost Einstein operator for $(2, 3, 5)$ distributions

Katja Sagerschnig, Travis Willse (2017)

Archivum Mathematicum

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For the geometry of oriented $(2, 3, 5)$ distributions $(M,)$ , which correspond to regular, normal parabolic geometries of type $(G_{2}, P)$ for a particular parabolic subgroup $P < G_{2}$ , we develop the corresponding tractor calculus and use it to analyze the first BGG operator $Θ_{0}$ associated to the $7$ -dimensional irreducible representation of $G_{2}$ . We give an explicit formula for the normal connection on the corresponding tractor bundle and use it to derive explicit expressions for this operator. We also show that solutions...

Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds

Neil Seshadri (2009)

Bulletin de la Société Mathématique de France

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To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$ , we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on $M \times (- 1, 0)$ . We consider the asymptotic expansion, in powers of a special defining function, of the volume of $M \times (- 1, 0)$ with respect to $g$ and prove that the log term coefficient is independent of $J$ (and any choice...

A variational characterization of condenser capacities in $C^{n}$ within a class of plurisubharmonic functions

Jerzy Kalina (1983)

Annales Polonici Mathematici

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