Showing papers in "Mathematische Annalen in 2021"

Log-concavity of volume and complex Monge–Ampère equations with prescribed singularity

Abstract: Let $$(X,\omega )$$ be a compact Kahler manifold. We prove the existence and uniqueness of solutions to complex Monge–Ampere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theorems in the context of big cohomology classes, however our results are new in the Kahler case as well. As an application we confirm a conjecture by Boucksom–Eyssidieux–Guedj–Zeriahi concerning log-concavity of the volume of closed positive (1, 1)-currents. Finally, we show that log-concavity of the volume in complex geometry corresponds to the Brunn–Minkowski inequality in convex geometry, pointing out a dictionary between our relative pluripotential theory and P-relative convex geometry. Applications related to stability and existence of csck metrics are treated elsewhere.

On the Rankin--Selberg problem

Abstract: In this paper, we solve the Rankin–Selberg problem. That is, we break the well known Rankin–Selberg’s bound on the error term of the second moment of Fourier coefficients of a $${\text {GL}}(2)$$ cusp form (both holomorphic and Maass), which remains its record since its birth for more than 80 years. We extend our method to deal with averages of coefficients of L-functions which can be factorized as a product of a degree one and a degree three L-functions.

Martingale optimal transport duality

Abstract: We obtain a dual representation of the Kantorovich functional defined for functions on the Skorokhod space using quotient sets. Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. These sets contain stochastic integrals defined pathwise and two such definitions starting with simple integrands are given. Another important ingredient of our analysis is a regularized version of Jakubowski’s S-topology on the Skorokhod space.

The uniqueness of inverse problems for a fractional equation with a single measurement

Abstract: This article is concerned with an inverse problem on simultaneously determining some unknown coefficients and/or an order of derivative in a multidimensional time-fractional evolution equation either in a Euclidean domain or on a Riemannian manifold. Based on a special choice of the Dirichlet boundary input, we prove the unique recovery of at most two out of four $$\varvec{x}$$ -dependent coefficients (possibly with an extra unknown fractional order) by a single measurement of the partial Neumann boundary output. Especially, both a vector-valued velocity field of a convection term and a density can also be uniquely determined. The key ingredient turns out to be the time-analyticity of the decomposed solution, which enables the construction of Dirichlet-to-Neumann maps in the frequency domain and thus the application of inverse spectral results.

Multiple normalized solutions for a Sobolev critical Schrödinger equation

Abstract: We study the existence of standing waves, of prescribed $$L^2$$ -norm (the mass), for the nonlinear Schrodinger equation with mixed power nonlinearities $$\begin{aligned} i \partial _t \phi + \Delta \phi + \mu \phi |\phi |^{q-2} + \phi |\phi |^{2^* - 2} = 0, \quad (t, x) \in \mathbb {R}\times \mathbb {R}^N, \end{aligned}$$ where $$N \ge 3$$ , $$\phi : \mathbb {R}\times \mathbb {R}^N \rightarrow \mathbb {C}$$ , $$\mu > 0$$ , $$2< q < 2 + 4/N $$ and $$2^* = 2N/(N-2)$$ is the critical Sobolev exponent. It was proved in Jeanjean et al. (Orbital stability of ground states for a Sobolev critical Schrodinger equation, 2020) that, for small mass, ground states exist and correspond to local minima of the associated Energy functional. It was also established that despite the nonlinearity is Sobolev critical, the set of ground states is orbitally stable. Here we prove that, when $$N \ge 4$$ , there also exist standing waves which are not ground states and are located at a mountain-pass level of the Energy functional. These solutions are unstable by blow-up in finite time. Our study is motivated by a question raised by Soave (J Funct Anal 279(6):108610, 2020).

Tensor categories of affine Lie algebras beyond admissible levels

Abstract: We show that if V is a vertex operator algebra such that all the irreducible ordinary V-modules are $$C_1$$ -cofinite and all the grading-restricted generalized Verma modules for V are of finite length, then the category of finite length generalized V-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra (resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie superalgebra) $$\mathfrak {g}$$ at level k and the category $$KL_k(\mathfrak {g})$$ of its finite length generalized modules, we discover several families of $$KL_k(\mathfrak {g})$$ at non-admissible levels k, having braided tensor category structures. In particular, $$KL_k(\mathfrak {g})$$ has a braided tensor category structure if the category of ordinary modules is semisimple or more generally if the category of ordinary modules is of finite length. We also prove the rigidity and determine the fusion rules of some categories $$KL_k(\mathfrak {g})$$ , including the category $$KL_{-1}(\mathfrak {sl}_n)$$ . Using these results, we construct a rigid tensor category structure on a full subcategory of $$KL_1(\mathfrak {sl}(n|m))$$ consisting of objects with semisimple Cartan subalgebra actions.

Vortex stretching and enhanced dissipation for the incompressible 3D Navier–Stokes equations

Abstract: We consider the 3D incompressible Navier–Stokes equations under the following $$2+\frac{1}{2}$$ -dimensional situation: small-scale horizontal vortex blob being stretched by large-scale, anti-parallel pairs of vertical vortex tubes. We prove enhanced dissipation induced by such vortex-stretching.

A Besov algebra calculus for generators of operator semigroups and related norm-estimates

Abstract: We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille–Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.

Korn and Poincaré-Korn inequalities for functions with a small jump set

Abstract: In this paper we prove a regularity and rigidity result for displacements in $$GSBD^p$$ , for every $$p>1$$ and any dimension $$n\ge 2$$ . We show that a displacement in $$GSBD^p$$ with a small jump set coincides with a $$W^{1,p}$$ function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincare-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in $$GSBD^p$$ .

Normalized solutions for a coupled Schrödinger system

Abstract: In the present paper, we prove the existence of solutions $$(\lambda _1,\lambda _2,u,v)\in \mathbb {R}^2\times H^1(\mathbb {R}^3,\mathbb {R}^2)$$ to systems of coupled Schrodinger equations $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1u=\mu _1 u^3+\beta uv^2\quad &{}\hbox {in}\;\mathbb {R}^3\\ -\Delta v+\lambda _2v=\mu _2 v^3+\beta u^2v\quad &{}\hbox {in}\;\mathbb {R}^3\\ u,v>0&{}\hbox {in}\;\mathbb {R}^3 \end{array}\right. } \end{aligned}$$ satisfying the normalization constraint $$ \int _{\mathbb {R}^3}u^2=a^2\quad \hbox {and}\;\int _{\mathbb {R}^3}v^2=b^2, $$ which appear in binary mixtures of Bose–Einstein condensates or in nonlinear optics. The parameters $$\mu _1,\mu _2,\beta >0$$ are prescribed as are the masses $$a,b>0$$ . The system has been considered mostly in the case of fixed frequencies $$\lambda _1,\lambda _2$$ . When the masses are prescribed, the standard approach to this problem is variational with $$\lambda _1,\lambda _2$$ appearing as Lagrange multipliers. Here we present a new approach based on the fixed point index in cones, bifurcation theory, and the continuation method. We obtain the existence of normalized solutions for any given $$a,b>0$$ for $$\beta $$ in a large range. We also have a result about the nonexistence of positive solutions which shows that our existence theorem is almost optimal. Especially, if $$\mu _1=\mu _2$$ we prove that normalized solutions exist for all $$\beta >0$$ and all $$a,b>0$$ .

Arithmetic hyperbolicity: automorphisms and persistence

Abstract: We show that if the automorphism group of a projective variety is torsion, then it is finite. Motivated by Lang’s conjecture on rational points of hyperbolic varieties, we use this to prove that a projective variety with only finitely many rational points has only finitely many automorphisms. Moreover, we investigate to what extent finiteness of S-integral points on a variety over a number field persists over finitely generated fields. To this end, we introduce the class of mildly bounded varieties and prove a general criterion for proving this persistence.

The unirationality of the moduli space of K3 surfaces of genus 22

Abstract: Using the connection discovered by Hassett between the Noether-Lefschetz moduli space $$\mathcal {C}_{42}$$ of special cubic fourfolds of discriminant 42 and the moduli space $$\mathcal {F}_{22}$$ of polarized K3 surfaces of genus 22, we show that the universal K3 surface over $$\mathcal {F}_{22}$$ is unirational.

Isometry groups of infinite-genus hyperbolic surfaces

Abstract: Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannian metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.

Sobolev estimates for solutions of the transport equation and ODE flows associated to non-Lipschitz drifts

Abstract: It is known, after Jabin (J Differ Equ 260(5):4739–4757, 2016) and Alberti et al. (Ann PDE 5(1):9, 2019), that ODE flows and solutions of the transport equation associated to Sobolev vector fields do not propagate Sobolev regularity, even of fractional order. In this paper, we improve the result at Clop and Jylha (J Differ Equ 266(8):4544–4567, 2019) and show that some kind of propagation of Sobolev regularity happens as soon as the gradient of the drift is exponentially integrable. We provide sharp Sobolev estimates and new examples. As an application of our main theorem, we generalize a regularity result for the 2D Euler equation obtained by Bahouri and Chemin in Bahouri and Chemin (Arch Ration Mech Anal 127(2):159–181, 1994).

Mass transference principle from rectangles to rectangles in Diophantine approximation

Abstract: The limsup sets defined by balls or defined by rectangles appear at the most fundamental level in Diophantine approximation: one follows from Dirichlet’s theorem, the other follows from Minkowski’s theorem. The metric theory for the former has been well studied, while the theory for the latter is rather incomplete, even in classic Diophantine approximation. This paper aims at setting up a general principle for the Hausdorff theory of limsup sets defined by rectangles. By introducing a notation called ubiquity for rectangles, a mass transference principle from rectangles to rectangles is presented, i.e., if a limsup set defined by a sequence of big rectangles has a full measure property, then this full measure property can be transferred to the full Hausdorff measure/dimension property for the limsup set defined by shrinking these big rectangles to smaller ones. Here the full measure property for the bigger limsup set goes with or without the assumption of ubiquity for rectangles. Together with the landmark work of Beresnevich and Velani (Ann Math (2) 164(3):971–992, 2006) where a transference principle from balls to balls is established, a coherent Hausdorff theory for metric Diophantine approximation is built. Besides completing the dimensional theory in simultaneous Diophantine approximation, linear forms, the dimensional theory for limsup sets defined by rectangles also underpins the dimensional theory in multiplicative Diophantine approximation where unexpected phenomenon occurs and the usually used methods fail to work.

Generalized traveling waves for time-dependent reaction???diffusion systems

Abstract: Traveling wave solutions in general time-dependent (including time-periodic) reaction–diffusion equations and systems of equations have attracted great attention in the last two decades. The aim of this paper is to study the propagation phenomenon in a general time-heterogeneous environment. More specifically, we investigate generalized traveling wave solutions for a two-component time-dependent non-cooperative reaction–diffusion system which has applications in epidemiology and ecology. Sufficient conditions on the existence and nonexistence of generalized traveling wave solutions are established. In the susceptible-infectious epidemic model setting, generalized traveling waves describe the spatio-temporal invasion of a disease into a totally susceptible population. In the context of predator–prey systems, the generalized traveling waves describe the spatial invasion of predators introduced into a new environment where the prey population is at its carrying capacity.

The action spectrum and $$C^0$$ C 0 symplectic topology

Abstract: Our first main result states that the spectral norm $$\gamma $$ on $$ \mathrm {Ham}(M, \omega ) $$ , introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the $$C^0$$ topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev-Shelukhin, we obtain the $$C^0$$ continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorphisms. Our second main result is related to the Arnold conjecture about fixed points of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian homeomorphism on any closed symplectic manifold of dimension greater than 2 having only one fixed point shows that the conjecture does not admit a direct generalization to the $$ C^0 $$ setting. However, in this paper we demonstrate that a reformulation of the conjecture in terms of fixed points as well as spectral invariants still holds for Hamiltonian homeomorphisms on symplectically aspherical manifolds.

Min–max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications

Abstract: For any smooth Riemannian metric on an $$(n+1)$$ -dimensional compact manifold with boundary $$(M,\partial M)$$ where $$3\le (n+1)\le 7$$ , we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min–max theory in the Almgren–Pitts setting. We apply our Morse index estimates to prove that for almost every (in the $$C^\infty $$ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If $$\partial M$$ is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.

Extension of multilinear fractional integral operators to linear operators on mixed-norm Lebesgue spaces

Abstract: In Kenig and Stein (Math Res Lett 6(1):1–15, 1999, https://doi.org/10.4310/MRL.1999.v6.n1.a1 ), the following type of multilinear fractional integral $$\begin{aligned} \int _{{\mathbb {R}}^{mn}} \frac{f_1(l_1(x_1,\ldots ,x_m,x))\cdots f_{m+1}(l_{m+1}(x_1,\ldots ,x_m,x))}{(|x_1|+\cdots +|x_m|)^{\lambda }} dx_1\ldots dx_m \end{aligned}$$ was studied, where $$l_i$$ are linear maps from $${\mathbb {R}}^{(m+1)n}$$ to $${\mathbb {R}}^n$$ satisfying certain conditions. They proved the boundedness of such multilinear fractional integral from $$L^{p_1}\times \cdots \times L^{p_{m+1}}$$ to $$L^q$$ when the indices satisfy the homogeneity condition. In this paper, we show that the above multilinear fractional integral extends to a linear operator for functions in the mixed-norm Lebesgue space $$L^{{\mathbf {p}}}$$ which contains $$L^{p_1}\times \cdots \times L^{p_{m+1}}$$ as a subset. Under less restrictions on the linear maps $$l_i$$ , we give a complete characterization of the indices $${\mathbf {p}}$$ , q and $$\lambda $$ for which such an operator is bounded from $$L^{{\mathbf {p}}}$$ to $$L^q$$ . And for $$m=1$$ or $$n=1$$ , we give necessary and sufficient conditions on $$(l_1, \ldots , l_{m+1})$$ , $${\mathbf {p}}=(p_1,\ldots , p_{m+1})$$ , q and $$\lambda $$ such that the operator is bounded.

Exponential sums with multiplicative coefficients without the Ramanujan conjecture

Abstract: We study the exponential sum involving multiplicative function f under milder conditions on the range of f, which generalizes the work of Montgomery and Vaughan. As an application, we prove cancellation in the sum of additively twisted coefficients of automorphic L-function on $$\text {GL}_m$$ $$(m\ge 4)$$ , uniformly in the additive character.

Affine Deligne–Lusztig varieties at infinite level

Abstract: We initiate the study of affine Deligne–Lusztig varieties with arbitrarily deep level structure for general reductive groups over local fields. We prove that for $${{\,\mathrm{GL}\,}}_n$$ and its inner forms, Lusztig’s semi-infinite Deligne–Lusztig construction is isomorphic to an affine Deligne–Lusztig variety at infinite level. We prove that their homology groups give geometric realizations of the local Langlands and Jacquet–Langlands correspondences in the setting that the Weil parameter is induced from a character of an unramified field extension. In particular, we resolve Lusztig’s 1979 conjecture in this setting for minimal admissible characters.

On certain subspaces of $\ell_p$ for $0<p\le 1$ and their applications to conditional quasi-greedy bases in $p$-Banach spaces

Abstract: We construct for each $$0

On the fill-in of nonnegative scalar curvature metrics

Abstract: In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data $$(\varSigma ,\gamma ,H)$$ . We prove that given a metric $$\gamma $$ on $${{\mathbf {S}}}^{n-1}$$ ( $$3\le n\le 7$$ ), $$({{\mathbf {S}}}^{n-1},\gamma ,H)$$ admits no fill-in of NNSC metrics provided the prescribed mean curvature H is large enough (Theorem 4). Moreover, we prove that if $$\gamma $$ is a positive scalar curvature (PSC) metric isotopic to the standard metric on $${{\mathbf {S}}}^{n-1}$$ , then the much weaker condition that the total mean curvature $$\int _{{{\mathbf {S}}}^{n-1}}H\,{{\mathrm {d}}}\mu _\gamma $$ is large enough rules out NNSC fill-ins, giving an partially affirmative answer to a conjecture by Gromov (Four lectures on scalar curvature, 2019, see P. 23). In the second part of this paper, we investigate the $$\theta $$ -invariant of Bartnik data and obtain some sufficient conditions for the existence of PSC fill-ins.

Ihara’s Lemma for Shimura curves over totally real fields via patching

Abstract: We prove Ihara’s lemma for the mod l cohomology of Shimura curves, localized at a maximal ideal of the Hecke algebra, under a large image hypothesis on the associated Galois representation. This was proved by Diamond and Taylor, for Shimura curves over $$\mathbb {Q}$$ , under various assumptions on l. Our method is totally different and can avoid these assumptions, at the cost of imposing the large image hypothesis. It uses the Taylor–Wiles method, as improved by Diamond and Kisin, and the geometry of integral models of Shimura curves at an auxiliary prime.

WDVV-type relations for disk Gromov–Witten invariants in dimension 6

Abstract: The first author’s previous work established Solomon’s WDVV-type relations for Welschinger’s invariant curve counts in real symplectic fourfolds by lifting geometric relations over possibly unorientable morphisms. We apply her framework to obtain WDVV-style relations for the disk invariants of real symplectic sixfolds with some symmetry, in particular confirming Alcolado’s prediction for $${\mathbb {P}}^3$$ and extending it to other spaces. These relations reduce the computation of Welschinger’s invariants of many real symplectic sixfolds to invariants in small degrees and provide lower bounds for counts of real rational curves with positive-dimensional insertions in some cases. In the case of $${\mathbb {P}}^3$$ , our lower bounds fit perfectly with Kollar’s vanishing results.

On Severi type inequalities

Abstract: We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities are related to some naturally defined birational invariants of the general fibers of the Albanese morphisms. As an application, we show that the volume of an irregular threefold of general type is at least $$\frac{3}{8}$$ . We also show that the volume of a smooth projective variety X of general type and of maximal Albanese dimension is at least $$2(\dim X)!$$ . Moreover, if $${{\,\mathrm{vol}\,}}(X)=2(\dim X)!$$ , the canonical model of X is a double cover of a principally polarized abelian variety $$(A, \Theta )$$ branched over some divisor $$D\in |2\Theta |$$ .

An improved result for Falconer’s distance set problem in even dimensions

Abstract: We show that if compact set $$E\subset \mathbb {R}^d$$ has Hausdorff dimension larger than $$\frac{d}{2}+\frac{1}{4}$$ , where $$d\ge 4$$ is an even integer, then the distance set of E has positive Lebesgue measure. This improves the previously best known result towards Falconer’s distance set conjecture in even dimensions.

Modularity and value distribution of quantum invariants of hyperbolic knots

Abstract: We obtain an exact modularity relation for the q-Pochhammer symbol. Using this formula, we show that Zagier’s modularity conjecture for a knot K essentially reduces to the arithmeticity conjecture for K. In particular, we show that Zagier’s conjecture holds for hyperbolic knots $$K e 7_2$$ with at most seven crossings. For $$K=4_1$$ , we also prove a complementary reciprocity formula which allows us to prove a law of large numbers for the values of the colored Jones polynomials at roots of unity. We conjecture a similar formula holds for all knots and we show that this is the case if one assumes a suitable version of Zagier’s conjecture.

A proof of the Khavinson conjecture

Abstract: We give a complete proof of the Khavinson conjecture which states that, for bounded harmonic functions on the unit ball of $$\mathbb {R}^n$$ , the sharp constants in the estimates for their radial derivatives and for their gradients coincide.

Homeomorphic extension of quasi-isometries for convex domains in $${\mathbb {C}}^d$$ C d and iteration theory

Abstract: We study the homeomorphic extension of biholomorphisms between convex domains in $${\mathbb {C}}^d$$ without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between the Gromov boundary and the topological boundaries of the domains and the dynamical properties of commuting 1-Lipschitz maps in Gromov hyperbolic spaces. This approach not only allows us to prove extensions for biholomorphisms, but for more general quasi-isometries between the domains endowed with their Kobayashi distances.