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

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

Similarity:

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 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 χ ( X , ω ) . This is done by...

On a Monge-Ampère type equation in the Cegrell class χ

Rafał Czyż (2010)

Annales Polonici Mathematici

Similarity:

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 ) = 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

Similarity:

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...

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

Similarity:

In this paper we consider Riemannian manifolds ( M n , g ) of dimension n 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,...

Concerning the energy class p for 0 < p < 1

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

Annales Polonici Mathematici

Similarity:

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

The gradient lemma

Urban Cegrell (2007)

Annales Polonici Mathematici

Similarity:

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

Similarity:

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 ) on the conformal compactification 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 ( X ¯ , X ¯ ) and the kernel of the Branson–Gover [3] differential operators ( L k , G k ) on the conformal infinity ( X ¯ , [ h 0 ] ) . We also relate the set of C n - 2 k + 1 ( Λ k ( X ¯ ) ) forms in the kernel of d + δ g ...

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

Zdeněk Dušek (2015)

Czechoslovak Mathematical Journal

Similarity:

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 ) 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

Similarity:

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...

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

Neil Seshadri (2009)

Bulletin de la Société Mathématique de France

Similarity:

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 × ( - 1 , 0 ) . We consider the asymptotic expansion, in powers of a special defining function, of the volume of M × ( - 1 , 0 ) with respect to g and prove that the log term coefficient is independent of J (and any choice...

Green functions, Segre numbers, and King’s formula

Mats Andersson, Elizabeth Wulcan (2014)

Annales de l’institut Fourier

Similarity:

Let 𝒥 be a coherent ideal sheaf on a complex manifold X with zero set Z , and let G be a plurisubharmonic function such that G = log | f | + 𝒪 ( 1 ) locally at Z , where f is a tuple of holomorphic functions that defines 𝒥 . We give a meaning to the Monge-Ampère products ( d d c G ) k for k = 0 , 1 , 2 , ... , and prove that the Lelong numbers of the currents M k 𝒥 : = 1 Z ( d d c G ) k at x coincide with the so-called Segre numbers of J at x , introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that M k 𝒥 satisfy a certain...

Two-dimensional curvature functionals with superquadratic growth

Ernst Kuwert, Tobias Lamm, Yuxiang Li (2015)

Journal of the European Mathematical Society

Similarity:

For two-dimensional, immersed closed surfaces f : Σ n , we study the curvature functionals p ( f ) and 𝒲 p ( f ) with integrands ( 1 + | A | 2 ) p / 2 and ( 1 + | H | 2 ) p / 2 , respectively. Here A is the second fundamental form, H is the mean curvature and we assume p > 2 . Our main result asserts that W 2 , p critical points are smooth in both cases. We also prove a compactness theorem for 𝒲 p -bounded sequences. In the case of p this is just Langer’s theorem [16], while for 𝒲 p we have to impose a bound for the Willmore energy strictly below 8 π as an additional...

η -Ricci Solitons on η -Einstein ( L C S ) n -Manifolds

Shyamal Kumar Hui, Debabrata Chakraborty (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Similarity:

The object of the present paper is to study η -Ricci solitons on η -Einstein ( L C S ) n -manifolds. It is shown that if ξ is a recurrent torse forming η -Ricci soliton on an η -Einstein ( L C S ) n -manifold then ξ is (i) concurrent and (ii) Killing vector field.

The Killing Tensors on an n -dimensional Manifold with S L ( n , ) -structure

Sergey E. Stepanov, Irina I. Tsyganok, Marina B. Khripunova (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Similarity:

In this paper we solve the problem of finding integrals of equations determining the Killing tensors on an n -dimensional differentiable manifold M endowed with an equiaffine S L ( n , ) -structure and discuss possible applications of obtained results in Riemannian geometry.

Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

Régis Monneau, Jean-Michel Roquejoffre, Violaine Roussier-Michon (2013)

Annales scientifiques de l'École Normale Supérieure

Similarity:

We construct travelling wave graphs of the form z = - c t + φ ( x ) , φ : x N - 1 φ ( x ) , N 2 , solutions to the N -dimensional forced mean curvature motion V n = - c 0 + κ ( c c 0 ) with prescribed asymptotics. For any 1 -homogeneous function φ , viscosity solution to the eikonal equation | D φ | = ( c / c 0 ) 2 - 1 , we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by  φ . We also describe φ in terms of a probability measure on  § N - 2 .

The CR Yamabe conjecture the case n = 1

Najoua Gamara (2001)

Journal of the European Mathematical Society

Similarity:

Let ( M , θ ) be a compact CR manifold of dimension 2 n + 1 with a contact form θ , and L = ( 2 + 2 / n ) Δ b + R its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact form θ ˜ on M conformal to θ which has a constant Webster curvature. This problem is equivalent to the existence of a function u such that L u = u 1 + 2 / n , u > 0 on M . D. Jerison and J. M. Lee solved the CR Yamabe problem in the case where n 2 and ( M , θ ) is not locally CR equivalent to the sphere S 2 n + 1 of 𝐂 n . In a join work with R. Yacoub, the CR Yamabe...

A geometric problem and the Hopf Lemma. I

Yan Yan Li, Louis Nirenberg (2006)

Journal of the European Mathematical Society

Similarity:

A classical result of A. D. Alexandrov states that a connected compact smooth n -dimensional manifold without boundary, embedded in n + 1 , and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane X n + 1 = const in case M satisfies: for any two points ( X ' , X n + 1 ) , ( X ' , X ^ n + 1 ) on M , with X n + 1 > X ^ n + 1 , the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for n = 1 ....