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Abstract: Motivated by Witten’s work, we propose a K-theoretic Verlinde/Grassmannian correspondence which relates the GL Verlinde numbers to the K-theoretic quasimap invariants of the Grassmannian. We recover these two types of invariants by imposing different stability conditions on the gauged linear sigma model associated with the Grassmannian. We construct two families of stability conditions connecting the two theories and prove two wall-crossing results. We confirm the Verlinde/Grassmannian correspondence in the rank two case. PubDate: 2022-05-14

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Abstract: Abstract We develop a min–max theory for asymptotically conical self-expanders of mean curvature flow. In particular, we show that given two distinct strictly stable self-expanders that are asymptotic to the same cone and bound a domain, there exists a new asymptotically conical self-expander trapped between the two. PubDate: 2022-04-11

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Abstract: Abstract The 2016 papers of Solomon and Tukachinsky use bounding chains in Fukaya’s \(A_{\infty }\) -algebras to define numerical disk counts relative to a Lagrangian under certain regularity assumptions on the moduli spaces of disks. We present a (self-contained) direct geometric analogue of their construction under weaker topological assumptions, extend it over arbitrary rings in the process, and sketch an extension without any assumptions over rings containing the rationals. This implements the intuitive suggestion represented by their drawing and Georgieva’s perspective. We also note a curious relation for the standard Gromov–Witten invariants readily deducible from their work. In a sequel, we use the geometric perspective of this paper to relate Solomon–Tukachinsky’s invariants to Welschinger’s open invariants of symplectic sixfolds, confirming their belief and Tian’s related expectation concerning Fukaya’s earlier construction. PubDate: 2022-02-04 DOI: 10.1007/s42543-021-00044-8

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Abstract: Abstract This paper proves that on any tamed closed almost complex four-manifold (M, J) whose dimension of J-anti-invariant cohomology is equal to the self-dual second Betti number minus one, there exists a new symplectic form compatible with the given almost complex structure J. In particular, if the self-dual second Betti number is one, we give an affirmative answer to a question of Donaldson for tamed closed almost complex four-manifolds. Our approach is along the lines used by Buchdahl to give a unified proof of the Kodaira conjecture. PubDate: 2022-02-03 DOI: 10.1007/s42543-021-00045-7

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Abstract: Abstract In this article, we present the concavity of the minimal \(L^2\) integrals related to multiplier ideals sheaves on Stein manifolds. As applications, we obtain a necessary condition for the concavity degenerating to linearity, a characterization for 1-dimensional case, and a characterization for the equality in 1-dimensional optimal \(L^{2}\) extension problem to hold. PubDate: 2022-01-22 DOI: 10.1007/s42543-021-00047-5

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Abstract: Abstract In this paper, we classify Möbius invariant differential operators of second order in two-dimensional Euclidean space, and establish a Liouville type theorem for general Möbius invariant elliptic equations. The equations are naturally associated with a continuous family of convex cones \(\Gamma _p\) in \(\mathbb R^2\) , with parameter \(p\in [1, 2]\) , joining the half plane \(\Gamma _1:=\{ (\lambda _1, \lambda _2):\lambda _1+\lambda _2>0\}\) and the first quadrant \(\Gamma _2:=\{ (\lambda _1, \lambda _2):\lambda _1, \lambda _2>0\}\) . Chen and C. M. Li established in 1991 a Liouville type theorem corresponding to \(\Gamma _1\) under an integrability assumption on the solution. The uniqueness result does not hold without this assumption. The Liouville type theorem we establish in this paper for \(\Gamma _p\) , \(1<p\le 2\) , does not require any additional assumption on the solution as for \(\Gamma _1\) . This is reminiscent of the Liouville type theorems in dimensions \(n\ge 3\) established by Caffarelli, Gidas and Spruck in 1989 and by A. B. Li and Y. Y. Li in 2003–2005, where no additional assumption was needed either. On the other hand, there is a striking new phenomena in dimension \(n=2\) that \(\Gamma _p\) for \(p=1\) is a sharp dividing line for such uniqueness result to hold without any further assumption on the solution. In dimensions \(n\ge 3\) , there is no such dividing line. PubDate: 2021-11-25 DOI: 10.1007/s42543-021-00043-9

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Abstract: Abstract Consider neutron transport equations in 3D convex domains with in-flow boundary. We mainly study the asymptotic limits as the Knudsen number \(\epsilon \rightarrow 0^+\) . Using Hilbert expansion, we rigorously justify that the solution of steady problem converges to that of Laplace’s equation, and the solution of unsteady problem converges to that of heat equation. This is the most difficult case of a long-term project on asymptotic analysis of kinetic equations in bounded domains. The proof relies on a detailed analysis on the boundary layer effect with geometric correction. The upshot of this paper is a novel boundary layer decomposition argument in 3D and \(L^2-L^{2m}-L^{\infty }\) bootstrapping method for time-dependent problem. PubDate: 2021-09-01 DOI: 10.1007/s42543-020-00032-4

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Abstract: Abstract On a complete noncompact Kähler manifold \(M^{n}\) (complex dimension) with nonnegative Ricci curvature and Euclidean volume growth, we prove that polynomial growth holomorphic functions of degree d has an dimension upper bound \(cd^{n}\) , where c depends only on n and the asymptotic volume ratio. Note that the power is sharp. PubDate: 2021-09-01 DOI: 10.1007/s42543-021-00034-w

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Abstract: Abstract We define and study cocycles on a Coxeter group in each degree generalizing the sign function. When the Coxeter group is a Weyl group, we explain how the degree three cocycle arises naturally from geometric representation theory. PubDate: 2021-09-01 DOI: 10.1007/s42543-020-00029-z

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Abstract: In this paper, we show that the elliptic cocenter of the Hecke algebra of a connected reductive p-adic group is contained in the rigid cocenter. As applications, we prove the trace Paley–Wiener theorem and the abstract Selberg principle for mod-l representations. PubDate: 2021-09-01 DOI: 10.1007/s42543-020-00027-1

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Abstract: Abstract We construct (modified) scattering operators for the Vlasov–Poisson system in three dimensions, mapping small asymptotic dynamics as \(t\rightarrow -\infty\) to asymptotic dynamics as \(t\rightarrow +\infty\) . The main novelty is the construction of modified wave operators, but we also obtain a new simple proof of modified scattering. Our analysis is guided by the Hamiltonian structure of the Vlasov–Poisson system. Via a pseudo-conformal inversion, we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function. PubDate: 2021-08-06 DOI: 10.1007/s42543-021-00041-x

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Abstract: Abstract The arithmetic version of the frequency transition conjecture for the almost Mathieu operators was recently proved by Jitomirskaya and Liu [34]. We give a new proof via reducibility theory and duality, which derives from the method developed in [22] (in fact it is a simplified version). This new proof is applicable to more general quasiperiodic operators. PubDate: 2021-08-05 DOI: 10.1007/s42543-021-00040-y

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Abstract: Abstract We prove the following result: if a \(\,\,\,\,\,{\mathbb {Q}}\,\,\,\,\,\) -Fano variety is uniformly K-stable, then it admits a Kähler–Einstein metric. This proves the uniform version of Yau–Tian–Donaldson conjecture for all (singular) Fano varieties with discrete automorphism groups. We achieve this by modifying Berman–Boucksom–Jonsson’s strategy in the smooth case with appropriate perturbative arguments. This perturbation approach depends on the valuative criterion and non-Archimedean estimates, and is motivated by our previous paper. PubDate: 2021-08-02 DOI: 10.1007/s42543-021-00039-5

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Abstract: Abstract In this paper, we show that for a connected compact Lie group to be acceptable, it is necessary and sufficient that its derived subgroup is isomorphic to a direct product of the groups \({\text {SU}}(n)\) , \({\text {Sp}}(n)\) , \({\text {SO}}(2n+1)\) , \({\text {G}}_2\) , \({\text {SO}}(4)\) . We show that there are invariant functions on \({\text {SO}}_{4}({\mathbb {C}})^{2}\) which are not generated by 1-argument invariants, though the group \({\text {SO}}_{4}({\mathbb {C}})\) is acceptable. PubDate: 2021-07-04 DOI: 10.1007/s42543-021-00038-6

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Abstract: Abstract We show the validity of the relative dlt MMP over \({\mathbb{Q}}\) -factorial threefolds in all characteristics \(p>0\) . As a corollary, we generalise many recent results to low characteristics including: \(W{\mathcal{O}}\) -rationality of klt singularities, inversion of adjunction, and normality of divisorial centres up to a universal homeomorphism. PubDate: 2021-06-18 DOI: 10.1007/s42543-021-00037-7

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Abstract: Abstract We prove that for a relatively hyperbolic group, there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of the group. Under natural assumptions, a similar result holds for the critical exponent of a cusp-uniform action of the group on a hyperbolic metric space. As a corollary, we obtain that the critical exponent of a torsion-free geometrically finite Kleinian group can be arbitrarily approximated by those of proper quotient groups. This resolves a question of Dal’bo–Peigné–Picaud–Sambusetti. Our approach is based on the study of Patterson–Sullivan measures on Bowditch boundary of a relatively hyperbolic group and gives a series of results on growth functions of balls and cones. PubDate: 2021-06-16 DOI: 10.1007/s42543-020-00033-3

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Abstract: Abstract We discuss a possible definition for “k-width” of a closed d-manifold \(M^d\) , and on embedding \(M^d \overset{e}{\hookrightarrow } \mathbb {R}^n\) , \(n > d \ge k\) , generalizing the classical notion of width of a knot. We show that for every 3-manifold 2-width \((M^3) \le 2\) but that there are embeddings \(e_i: T^3 \hookrightarrow \mathbb {R}^4\) with 2-width \((e_i) \rightarrow \infty \) . We explain how the divergence of 2-width of embeddings offers a tool to which might prove the Goeritz groups \(G_g\) infinitely generated for \(g \ge 4\) . Finally we construct a homomorphism \(\theta _g: G_g \rightarrow \mathrm {MCG}(\underset{g}{\#} S^2 \times S^2)\) , suggesting a potential application of 2-width to 4D mapping class groups. PubDate: 2021-05-15 DOI: 10.1007/s42543-021-00035-9

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Abstract: Abstract We show the rigidity of the hexagonal Delaunay triangulated plane under Luo’s PL conformality. As a consequence, we obtain a rigidity theorem for a particular type of locally finite convex ideal hyperbolic polyhedra. PubDate: 2021-04-26 DOI: 10.1007/s42543-021-00036-8

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Abstract: Abstract In this paper, we use the Sobolev type inequality in Wang et al. (Moser–Trudinger inequality for the complex Monge–Ampère equation, arXiv:2003.06056v1 (2020)) to establish the uniform estimate and the Hölder continuity for solutions to the complex Monge–Ampère equation with the right-hand side in \(L^p\) for any given \(p>1\) . Our proof uses various PDE techniques but not the pluripotential theory. PubDate: 2021-03-01 DOI: 10.1007/s42543-020-00025-3

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Abstract: Abstract A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space. On the other hand, we provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green’s function and the positive mass theorem play essential roles in certain cases. PubDate: 2021-03-01 DOI: 10.1007/s42543-020-00028-0