Mostrando recursos 1 - 14 de 14

  1. Index to Volume 107

  2. An extension of a theorem of Wu–Yau

    Tosatti, Valentino; Yang, Xiaokui
    We show that a compact Kähler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the projective case by Wu–Yau.

  3. Isoperimetric inequality for the third eigenvalue of the Laplace–Beltrami operator on $\mathbb{S}^2$

    Nadirashvili, Nikolai; Sire, Yannick
    We prove Hersch’s type isoperimetric inequality for the third positive eigenvalue on $\mathbb{S}^2$. Our method builds on the theory we developed to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.

  4. Barycenter technique and the Riemann mapping problem of Cherrier–Escobar

    Mayer, Martin; Ndiaye, Cheikh Birahim
    We solve in the affirmative the remaining cases of the Riemann mapping problem of Cherrier–Escobar [35][38] first raised by Cherrier [35] in 1984. Indeed, performing a suitable scheme of the barycenter technique of Bahri–Coron [14] via the Almaraz–Chen’s bubbles [3][34], we completely solve all the cases left open after the work of Chen [34]. Hence, combining our work with the ones of Almaraz [2], Chen [34], Cherrier [35], Escobar [38][40] and Marques [55][56], we have that every compact Riemannian manifold with boundary, of dimension greater or equal than three, and with finite Sobolev quotient, carries a conformal scalar flat metric...

  5. A twisted nonabelian Hodge correspondence

    García-Raboso, Alberto
    We prove an extension of the nonabelian Hodge theorem [Sim92] in which the underlying objects are twisted torsors over a smooth complex projective variety. In the prototypical case of $GL_n$-torsors, one side of this correspondence consists of vector bundles equipped with an action of a sheaf of twisted differential operators in the sense of Beĭlinson and Bernstein [BB93]; on the other side, we endow them with appropriately defined twisted Higgs data. ¶ The proof we present here is formal, in the sense that we do not delve into the analysis involved in the classical nonabelian Hodge correspondence. Instead, we use...

  6. Degenerations of Abelian differentials

    Chen, Dawei
    Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear series, we define twisted canonical divisors on pointed nodal curves to study degenerate differentials, give dimension bounds for their moduli spaces, and establish smoothability criteria. As applications, we show that the spin parity of holomorphic and meromorphic differentials extends to distinguish twisted canonical divisors in the locus of stable pointed curves of pseudocompact type. We also justify whether zeros and poles on general curves in a stratum of differentials can be Weierstrass points. Moreover, we classify twisted canonical divisors...

  7. The nonequivariant coherent-constructible correspondence for toric surfaces

    Kuwagaki, Tatsuki
    We prove the nonequivariant coherent-constructible correspondence conjectured by Fang–Liu–Treumann–Zaslow in the case of toric surfaces. Our proof is based on describing a semi-orthogonal decomposition of the constructible side under toric point blow-up and comparing it with Orlov’s theorem.

  8. Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II

    Donaldson, Simon; Sun, Song
    We study Gromov–Hausdorff limits of Kähler–Einstein manifolds, in particular, their singularities, and connections with algebraic geometry. This is a continuation of our previous work.

  9. Transverse singularities of minimal two-valued graphs in arbitrary codimension

    Becker-Kahn, Spencer T.
    We prove some epsilon regularity results for $n$-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one $n$-dimensional planes, and in a neighbourhood of any point at which at which at least one tangent cone is equal to a union of four distinct multiplicity one $n$-dimensional half-planes that meet along an $(n-1)$-dimensional axis. The key ingredient is a new Excess Improvement Lemma obtained via a blow-up method (inspired...

  10. Convolution of valuations on manifolds

    Alesker, Semyon; Bernig, Andreas
    We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and generalized valuations on $M$ are modules over the algebra of compactly supported generalized valuations on $G$ satisfying some technical condition of tameness. ¶ The case of a vector space acting on itself is studied in detail. We prove explicit formulas in this case and show that the new convolution is an extension of the convolution on...

  11. Regularization and minimization of codimension-one Haefliger structures

    Meigniez, Gaël
    On compact manifolds of dimensions $4$ and more, we give a proof of Thurston’s existence theorem for foliations of codimension one; that is, they satisfy some $h$-principle in the sense of Gromov. Our proof is an explicit construction not using the Mather homology equivalence. Moreover, the produced foliations are minimal, that is, all leaves are dense. In particular, there exist minimal, $C^{\infty}$, codimension-one foliations on every closed connected manifold of dimension at least $4$ whose Euler characteristic is zero.

  12. On the singularities of the Szegő projections on lower energy forms

    Hsiao, Chin-Yu; Marinescu, George
    Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1 , n \geqslant 2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian on $(0, q)$-forms. We show that the spectral function of $\Box^{(q)}_{b}$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$, then the spectrum of $\Box^{(q)}_{b}$ in $] 0, \infty [$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated...

  13. Convergence of the $J$-flow on toric manifolds

    Collins, Tristan C.; Székelyhidi, Gábor
    We show that on a Kähler manifold whether the $J$-flow converges or not is independent of the chosen background metric in its Kähler class. On toric manifolds we give a numerical characterization of when the $J$-flow converges, verifying a conjecture in [19] (M. Lejmi and G. Székelyhidi, “The $J$-flow and stability”) in this case. We also strengthen existing results on more general inverse $\sigma_k$ equations on Kähler manifolds.

  14. On static three-manifolds with positive scalar curvature

    Ambrozio, Lucas
    We compute a Bochner type formula for static three-manifolds and deduce some applications in the case of positive scalar curvature. We also explain in details the known general construction of the (Riemannian) Einstein $(n + 1)$-manifold associated to a maximal domain of a static $n$-manifold where the static potential is positive. There are examples where this construction inevitably produces an Einstein metric with conical singularities along a codimension-two submanifold. By proving versions of classical results for Einstein four-manifolds for the singular spaces thus obtained, we deduce some classification results for compact static three-manifolds with positive scalar curvature.

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