Abstract: In this note, we first give some constructions of torsion-free \(G_2\) -structures on some topological product manifolds. Then we provide a sufficient condition of 3-Calabi–Yau fibrations for \(G_2\) -manifolds. Next we study the Gukov–Yau–Zaslow horizontal lifting for hyperKähler firbations of \(G_2\) -manifolds, and discuss when the Gukov-Yau-Zaslow metric on this fibration is a \(G_2\) -metric. PubDate: 2019-03-22
Abstract: On a two-dimensional compact Riemannian manifold with boundary, we prove that the first nonzero Steklov eigenvalue is nondecreasing along the unnormalized geodesic curvature flow if the initial metric has positive geodesic curvature and vanishing Gaussian curvature. Using the normalized geodesic curvature flow, we also obtain some estimate for the first nonzero Steklov eigenvalue. On the other hand, we prove that the compact soliton of the geodesic curvature flow must be the trivial one. PubDate: 2019-03-21
Abstract: Let X be a pointed CW-complex. The generalized conjecture on spherical classes states that, the Hurewicz homomorphism \(H: \pi _{*}(Q_0 X) \rightarrow H_{*}(Q_0 X)\) vanishes on classes of \(\pi _* (Q_0 X)\) of Adams filtration greater than 2. Let \( \varphi _s: Ext _{\mathcal {A}}^{s}(\widetilde{H}^*(X), \mathbb {F}_2) \rightarrow {(\mathbb {F}_2 \otimes _{{\mathcal {A}}}R_s\widetilde{H}^*(X))}^* \) denote the sth Lannes–Zarati homomorphism for the unstable \({\mathcal {A}}\) -module \(\widetilde{H}^*(X)\) . This homomorphism corresponds to an associated graded of the Hurewicz map. An algebraic version of the conjecture states that the sth Lannes–Zarati homomorphism vanishes in any positive stem for \(s>2\) and any CW-complex X. We construct a chain level representation for the Lannes–Zarati homomorphism by means of modular invariant theory. We show the commutativity of the Lannes–Zarati homomorphism and the squaring operation. The second Lannes–Zarati homomorphism for \(\mathbb {R}\mathbb {P}^{\infty }\) vanishes in positive stems, while the first Lannes-Zatati homomorphism for any space is basically non-zero. We prove the algebraic conjecture for \(\mathbb {R}\mathbb {P}^{\infty }\) and \(\mathbb {R}\mathbb {P}^{n}\) with \(s=3\) , 4. We discuss the relation between the Lannes–Zarati homomorphisms for \(\mathbb {R}\mathbb {P}^{\infty }\) and \(S^0\) . Consequently, the algebraic conjecture for \(X=S^0\) is re-proved with \(s=3\) , 4, 5. PubDate: 2019-03-19
Abstract: In this paper, we prove that the concavity of Rényi entropy power of positive solutions to the parabolic p-Laplace equations on compact Riemannian manifold with nonnegative Ricci curvature. As applications, we derive the improved \(L^p\) -Gagliardo-Nirenberg inequalities. PubDate: 2019-03-16
Abstract: In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in \(\mathbb {R}^{n+1}\) , giving some conditions under which a translating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in \(R^{n+k}\) , namely, if the \(L^n\) norm of the second fundamental form of the soliton is small enough, then it is a hyperplane. PubDate: 2019-03-15
Abstract: We exhibit a generating function of spin Hurwitz numbers analogous to (disconnected) double Hurwitz numbers that is a tau function of the two-component BKP (2-BKP) hierarchy and is a square root of a tau function of the two-component KP (2-KP) hierarchy defined by related Hurwitz numbers. PubDate: 2019-03-15
Abstract: Let \({\widetilde{k}}\) be a fixed cubic field, F a quadratic field and \(L=\widetilde{k}\cdot F\) . In this paper and its companion paper, we determine the density of more or less the ratio of the residues of the Dedekind zeta functions of L, F where F runs through quadratic fields. PubDate: 2019-03-14
Abstract: We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead of using cohomology, we deal with the Chow motives. More precisely, we prove that for any smooth proper two-dimensional orbifold with projective coarse moduli space, there is an isomorphism of algebra objects, in the category of complex Chow motives, between the motive of the minimal resolution and the orbifold motive. In particular, the complex Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the minimal resolution is isomorphic to the complex orbifold Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the orbifold surface. This confirms the two-dimensional Motivic Crepant Resolution Conjecture. PubDate: 2019-03-01
Abstract: We estimate the number of homotopy types of path-components of the mapping spaces \(M(\mathbb {S}^m,\mathbb {F}P^n)\) from the m-sphere \(\mathbb {S}^m\) to the projective space \(\mathbb {F}P^n\) for \(\mathbb {F}\) being the real numbers \(\mathbb {R}\) , the complex numbers \(\mathbb {C}\) , or the skew algebra \(\mathbb {H}\) of quaternions. Then, the homotopy types of path-components of the mapping spaces \(M(E\Sigma ^m,\mathbb {F}P^n)\) for the suspension \(E\Sigma ^m\) of a homology m-sphere \(\Sigma ^m\) are studied as well. PubDate: 2019-03-01
Abstract: We establish a local monotonicity formula for mean curvature flow into a curved space whose metric is also permitted to evolve simultaneously with the flow, extending the work of Ecker (Ann Math (2) 154(2):503–525, 2001), Huisken (J Differ Geom 31(1):285–299, 1990), Lott (Commun Math Phys 313(2):517–533, 2012), Magni, Mantegazza and Tsatis (J Evol Equ 13(3):561–576, 2013) and Ecker et al. (J Reine Angew Math 616:89–130, 2008). This formula gives rise to a monotonicity inequality in the case where the target manifold’s geometry is suitably controlled, as well as in the case of a gradient shrinking Ricci soliton. Along the way, we establish suitable local energy inequalities to deduce the finiteness of the local monotone quantity. PubDate: 2019-03-01
Abstract: We prove that the group of isometries of a metric measure space that satisfies the Riemannian curvature condition, \(RCD^*(K,N),\) is a Lie group. We obtain an optimal upper bound on the dimension of this group, and classify the spaces where this maximal dimension is attained. PubDate: 2019-03-01
Abstract: In this paper we prove the existence of a positive solution of the nonlinear and nonlocal elliptic equation in \(\mathbb {R}^n\) $$\begin{aligned} (-\Delta )^s u =\varepsilon h u^q+u^{2_s^*-1} \end{aligned}$$ in the convex case \(1\le q<2_s^*-1\) , where \( 2_s^*={2n}/({n-2s}) \) is the critical fractional Sobolev exponent, \((-\Delta )^s\) is the fractional Laplace operator, \(\varepsilon \) is a small parameter and h is a given bounded, integrable function. The problem has a variational structure and we prove the existence of a solution by using the classical Mountain-Pass Theorem. We work here with the harmonic extension of the fractional Laplacian, which allows us to deal with a weighted (but possibly degenerate) local operator, rather than with a nonlocal energy. In order to overcome the loss of compactness induced by the critical power we use a Concentration-Compactness principle. Moreover, a finer analysis of the geometry of the energy functional is needed in this convex case with respect to the concave–convex case studied in Dipierro et al. (Fractional elliptic problems with critical growth in the whole of \(\mathbb {R}^n\) . Lecture Notes Scuola Normale Superiore di Pisa, vol 15. Springer, Berlin, 2017). PubDate: 2019-03-01
Abstract: We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential and a Caratheodory reaction term which is resonant both at zero and \(\pm \infty \) . Using the Lyapunov–Schmidt reduction method and critical groups (Morse theory), we show that the problem has at least two nontrivial smooth solutions. PubDate: 2019-03-01
Abstract: We study invariant metrics on Ledger–Obata spaces \(F^m/{\text {diag}}(F)\) . We give the classification and an explicit construction of all naturally reductive metrics, and also show that in the case \(m=3\) , any invariant metric is naturally reductive. We prove that a Ledger–Obata space is a geodesic orbit space if and only if the metric is naturally reductive. We then show that a Ledger–Obata space is reducible if and only if it is isometric to the product of Ledger–Obata spaces (and give an effective method of recognising reducible metrics), and that the full connected isometry group of an irreducible Ledger–Obata space \(F^m/{\text {diag}}(F)\) is \(F^m\) . We deduce that a Ledger–Obata space is a geodesic orbit manifold if and only if it is the product of naturally reductive Ledger–Obata spaces. PubDate: 2019-03-01
Abstract: In this paper, we study the Gieseker moduli space \(\mathcal {M}_{1,1}^{4,3}\) of minimal surfaces with \(p_g=q=1, K^2=4\) and genus 3 Albanese fibration. Under the assumption that direct image of the canonical sheaf under the Albanese map is decomposable, we find two irreducible components of \(\mathcal {M}_{1,1}^{4,3}\) , one of dimension 5 and the other of dimension 4. PubDate: 2019-03-01
Abstract: Strata of k-differentials on smooth curves parameterize sections of the k-th power of the canonical bundle with prescribed orders of zeros and poles. Define the tautological ring of the projectivized strata using the \(\kappa \) and \(\psi \) classes of moduli spaces of pointed smooth curves along with the class \(\eta = \mathcal O(-1)\) of the Hodge bundle. We show that if there is no pole of order k, then the tautological ring is generated by \(\eta \) only, and otherwise it is generated by the \(\psi \) classes corresponding to the poles of order k. PubDate: 2019-03-01
Authors:Nathan Ilten; Marni Mishna; Charlotte Trainor Abstract: The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and Süß to a correspondence between Gorenstein Fano complexity-one T-varieties and Fano divisorial polytopes. Motivated by the finiteness of reflexive polytopes in fixed dimension, we show that over a fixed base polytope, there are only finitely many Fano divisorial polytopes, up to equivalence. We classify two-dimensional Fano divisorial polytopes, recovering Huggenberger’s classification of Gorenstein del Pezzo \(\mathbb {K}^*\) -surfaces. Furthermore, we show that any three-dimensional Fano divisorial polytope is equivalent to one involving only eight functions. PubDate: 2018-05-14 DOI: 10.1007/s00229-018-1036-x
Authors:Stella Piro-Vernier; Francesco Ragnedda; Vincenzo Vespri Abstract: In this note we show the Hölder regularity for bounded solutions to a class of anisotropic elliptic operators. This result is the dual of the one proved by Liskevich and Skrypnik (Nonlinear Anal 71:1699–1708, 2009). PubDate: 2018-05-02 DOI: 10.1007/s00229-018-1034-z
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