Supercritical Conformal Metrics on Surfaces with Conical Singularities

Bartolucci, Daniele; Marchis, Francesca De; Malchiodi, Andrea

doi:10.1093/imrn/rnq285

Cited by 87 publications

(143 citation statements)

References 33 publications

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“…By arguing as in [1,2], we find that J B2,ρ and J G2,ρ are Morse functionals for a generic choice of initial data:…”

Section: Lemma 24mentioning

confidence: 79%

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B 2 and G2 Toda systems on compact surfaces: A variational approach

Battaglia

2017

Journal of Mathematical Physics

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We consider the B2 and G2 Toda systems on a compact surface (Σ, g)where A = (aij) = 2 −1 −2 2 or 2 −1 −3 2 and hi ∈ C ∞ >0 (Σ), ρi ∈ R>0 are given. We attack the problem using variational techniques, following the previous work [4] concerning the A2 Toda system, namely the case A = 2 −1 −1 2 . We get existence and multiplicity of solutions as long as χ(Σ) ≤ 0 and ρ1, ρ2 ∈ 4πN. We also extend some of the results to the case of general systems. IntroductionLet (Σ, g) be a closed Riemann surface with surface area equal to 1. The B 2 and G 2 Toda systems are respectively the following systems of PDEs on Σ:Here, −∆ = −∆ g is the Laplace-Beltrami operator, ρ 1 , ρ 2 are positive parameters and h 1 , h 2 are positive smooth functions on Σ.Such systems are particularly interesting because their matrices of coefficients−3 2 * Sapienza Università di Roma, Dipartimento di Matematica, Piazzale Aldo Moro 5, 00185 Romabattaglia@mat.uniroma1.it 1 are the Cartan matrices of the special orthonormal group SO(5) and of the symplectic group Sp(4), respectively. Together with A 2 = 2 −1 −1 2 , corresponding to SU (3), these are the only 2-dimensional Cartan matrices. Such Toda systems have important applications in both algebraic geometry and mathematical physics. In geometry, they appear in the study of complex holomorphic curves (see [8,13,7,11]); in physics, they arise in non-Abelian gauge field theory (see [19,18,13,12,14]).Both (1) and (2) are variational problems. In fact, solutions are respectively critical points of the following energy functionals:logˆΣ h 1 e u1 dV g −ˆΣ u 1 dV g − ρ 2 2 logˆΣ h 2 e u2 dV g −ˆΣ u 2 dV g ; (3) J G2,ρ (u) :=ˆΣ Q G2 (u)dV g − ρ 1 logˆΣ h 1 e u1 dV g −ˆΣ u 1 dV g − ρ 2 3 logˆΣ h 2 e u2 dV g −ˆΣ u 2 dV g . (4)

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“…By arguing as in [1,2], we find that J B2,ρ and J G2,ρ are Morse functionals for a generic choice of initial data:…”

Section: Lemma 24mentioning

confidence: 79%

“…In fact, as was done in [1,2], if Σ's genus equals g > 1 we can take as γ 1 , γ 2 not just two circles but two wedge sums of g circles each. This yields higher-dimensional homology groups for low sub-levels which in turn gives, through Morse theory, a higher number of solutions.…”

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confidence: 99%

B 2 and G2 Toda systems on compact surfaces: A variational approach

Battaglia

2017

Journal of Mathematical Physics

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“…Under the hypotheses K > 0 and α j > 0, Bartolucci and Tarantello, [5], proved a concentration-compactness result which implies that blow-up can occur only if λ belongs to the following discrete set of values Besides, they also proved an existence result for (1.9) on surfaces with positive genus and λ ∈ (8π, 16π)\Γ(α m ), generalized by Bartolucci, De Marchis and Malchiodi in [2], obtaining solvability for any λ ∈ (8π, +∞) \ Γ(α m ).…”

Section: 2)mentioning

confidence: 96%

Existence and non existence results for the singular Nirenberg problem

Marchis

López-Soriano

2016

Calc. Var.

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ABSTRACT. In this paper we study the problem, posed by Troyanov in [47], of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a nonlinear PDE with exponential type non-linearity admitting a variational structure. In particular, we are concerned with the case where the prescribed function K changes sign. When the surface is the standard sphere, namely for the singular Nirenberg problem, we give sufficient conditions on K, concerning mainly the regularity of its nodal line and the topology of its positive nodal region, to be the Gaussian curvature of a conformal metric with assigned conical singularities.Besides, we find a class of functions on S 2 which do not verify our conditions and which can not be realized as the Gaussian curvature of any conformal metric with one conical singularity. This shows that our result is somehow sharp.

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“…The existence and uniqueness (or more likely, multiplicity) theory when any of the cone angles are greater than 2π is much less well understood. We refer to the work of Malchiodi and his collaborators [5,8,20] and Mondello and Panov [28] for the best current results in these directions.…”

Section: Vi-1mentioning

confidence: 99%

Sharp parabolic regularity and geometric flows on singular spaces

Mazzeo¹

2016

Journées Équations Aux Dérivées Partielles

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This is a brief survey about regularity expansions for solutions of elliptic and parabolic problems on spaces with conic singularities. The results themselves are closely related to classical results about elliptic boundary problems, and analogues of these are expected to hold on quite general stratified spaces with incomplete iterated edge metrics. The emphasis here is on the interpretation and application of these expansions to geometric problems.

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Supercritical Conformal Metrics on Surfaces with Conical Singularities

Cited by 87 publications

References 33 publications

B 2 and G2 Toda systems on compact surfaces: A variational approach

B 2 and G2 Toda systems on compact surfaces: A variational approach

Existence and non existence results for the singular Nirenberg problem

Sharp parabolic regularity and geometric flows on singular spaces

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