2019年度秋季総合分科会(於:金沢大学)

SUMMARY: Caloric morphism is the transformation which preserves solutions of the heat equation. On Euclidean spaces, Appell transformation is the typical and essential example.

In this talk, we introduce the notion of caloric morphism on Euclidean domains and give several characterizations of caloric morphism. As a result, we can determine caloric morphisms explicitly under some conditions. The Schwarzian derivative and its related derivatives appear in the process of determination.

Next, we generalize the notion of caloric morphism to Riemannian manifolds and give a characterization theorem.

Finally, we generalize caloric mophism further to semi-riemannian manifolds. This generalization reveals which property of caloric morphism depends on the positivity of the Laplacian.

SUMMARY: The technique of using Newton polyhedra has many significant applications in singularity theory. In this talk, we discuss some important subjects in several complex variables by using Newton polyhedra.

In the strictly pseudoconvex case, as is well known, there exists local holomorphic coordinates on which the boundary can be clearly expressed. This fact plays useful roles in various analyses on strictly pseudoconvex domains; for example, construction of peak functions, boundary behaviors of the Bergman kernel and Szegö kernel, boundary behavior of squeezing functions. On the other hand, in the weakly pseudoconvex case, a serious problem is understanding what kinds of coordinates are appropriate for a given analytical issue and how to express the boundary on these coordinates. We introduce some local holomorphic coordinates through properties of the Newton polyhedron associated to the boundary and precisely investigate the two issues: determination of the D’Angelo type and boundary behavior of the Bergman kernel. We give quantitative results for these issues from simple geometrical information of the respective Newton polyhadron. Note that the above two issues can be considered as those analogous to determination of the Łojasiewicz exponent and behavior of oscillatory integrals.

SUMMARY: In this talk, as a direct application of Q. Guan’s result on the conjugate analytic Hardy \(H_2\) norm we will derive a new type isoperimetric inequality for Dirichlet integrals of analytic functions.

SUMMARY: In this talk, we discuss the weighted \(m\)-polyharmonic Bergman space on the unit ball. We will give the estimate for the reproducing kernel of the orthogonal complement of the weighted \((m-1)\)-polyharmonic Bergman space in the weighted \(m\)-polyharmonic Bergman space.

SUMMARY: We consider a weighted version of Bergman type spaces for iterated parabolic operators of fractional order on the upper half space. First we verify reproducing properties for polyparabolic Bergman functions. Next we discuss some properties of polyparabolic Bergman space, for example, the completeness, the boundedness of point evaluations and norm inequalities. Finally we make a remark on the relation with polyharmonic Bergman spaces.

SUMMARY: The author developed the theory of dimensioned numbers, which is suitable for representing iterated power functions and iterated exponential functions. We explain how to construct dimensioned number solutions to an iterative functional equation, and discuss its real-analyticity.

SUMMARY: Let \(R\) be an open Riemann surface of genus \(g\, (0 < g < \infty )\) and \(\chi = \{A_j, B_j \}_{j=1}^g\) be a fixed canonical homology basis of \(R\) modulo dividing cycles. Suppose that a complex \(g\)-vector \(\mathbf c\) is given and consider any hydrodynamic differential \(\phi \) on \((R, \chi )\) whose \(A\)-period vector is \(\mathbf c\). We first show an identity which gives the \(B\)-period vector of \(\phi \). As an application we characterize the Riemann period matrix of a hydrodynamic closing. We also define a new type of span and show its geometric meaning.

SUMMARY: We prove that for any infinite collection of quadratic Julia sets, there exists a transcendental entire function whose Julia set contains quasiconformal copies of the given quadratic Julia sets. In order to prove the result, we construct a quasiregular map with required dynamics and employ the quasiconformal surgery to obtain the desired transcendental entire function.

SUMMARY: We introduce a coordinate system to the SL(2,C)-representation space of twice punctured torus groups to find some hyperbolic 3-manifolds which fibers over circle.

SUMMARY: For a hyperbolic surface of genus g, the maximal number of pairwise disjoint simple closed geodesics is \(3g-3\). We can also consider the maximal numbers of such geodesics for translation surfaces. A translation surface is a surface together with a singular Euclidean metric. A closed geodesic on a translation surface is either a union of segments connecting singularities or a geodesic without singularities. In the case of genus 2, the maximal numbers of pairwise disjoint and non-homotopic geodesics without singularities is studied by Nguyen. We give the maximal numbers of such geodesics for hyperelliptic translation surfaces of genus \(g\).

SUMMARY: In this talk, we will introduce the division by zero calculus in multiply dimensions in order to show some wide and new open problems as we see from one dimensional case.

SUMMARY: In this talk, we consider some properties of operators from the Hardy space \(H^{\infty }(B_X)\) to the \(\alpha \)-Bloch space on a finite dimensional bounded symmetric domain.

SUMMARY: Let \(\mathbb {B}_X\) be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*-triple \(X=(\mathbb {C}^n, \| \cdot \|_X)\). In this talk, we give a definition of \(\alpha \)-Bloch mappings on \(\mathbb {B}_X\) which is a generalization of \(\alpha \)-Bloch functions on the unit disc in \(\mathbb {C}\). This definition is new in the case of the Euclidean unit ball \(\mathbb {B}^n\) in \(\mathbb {C}^n\). We generalize Bonk’s distortion theorem to \(\alpha \)-Bloch mappings on \(\mathbb {B}_X\). As an application, we give a lower bound of the Bloch constant for \(\alpha \)-Bloch mappings on \(\mathbb {B}_X\).

SUMMARY: Let \(\mathbb {B}_X\) be a bounded symmetric domain realized as the open unit ball \(\mathbb {B}_X\) of a finite dimensional JB*-triple \(X\). In this talk, we continue the work related to \(\alpha \)-Bloch mappings on \(\mathbb {B}_X\). We first show that \(\alpha \)-Bloch spaces on \(\mathbb {B}_X\) are complex Banach spaces. Next, we give sufficient conditions for the composition operator from the \(\alpha \)-Bloch space into the \(\beta \)-Bloch space to be bounded or compact. In the case that the \(\alpha \)-Bloch space is a Bloch space, then these conditions are also necessary.

SUMMARY: In this talk, we will generalize the Bloch-type spaces and the little Bloch-type spaces to the open unit ball \(\mathbb {B}\) of a general infinite dimensional complex Banach space by using the radial derivative. Next, we define an extended Cesàro operator \(T_{\varphi }\) with holomorphic symbol \(\varphi \) and characterize those \(\varphi \) for which \(T_{\varphi }\) is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those \(\varphi \) for which \(T_{\varphi }\) is compact between the Bloch-type spaces and the little Bloch-type spaces under some additional assumption on the symbol \(\varphi \). When \(\mathbb {B}\) is the open unit ball of a finite dimensional complex Banach space \(X\), this additional assumption is automatically satisfied.

SUMMARY: We study a relation among the radius of the incircle of a convex polygon, its area and its sides length.

SUMMARY: The inversion problem of the hyperelliptic integrals of genus 2 is important in many fields such as computation of the conformal mapping of polygons and construction of exact solutions of the geodesic equations in physics. Grant gave a function which solves the inversion problem in terms of the two-dimensional sigma function. In this talk, we derive differential equations satisfied by the function and series expansion of the function. When the curves of genus 2 deform to elliptic curves, we show that the function transforms into the Weierstrass elliptic function.

SUMMARY: The torsion module of Kähler differential forms is considered. Relations between logarithmic differential forms and logarithmic vector fields are investigated. As an application, an effective method is proposed for computing torsion differential forms associated with a hypersurface with an isolated singularity. The main ingredients of the proposed method are logarithmic vector fields and local cohomology.

SUMMARY: A system of natural numbers \(({\bold a}; h) = (a_1, ... , a_{d+1};h)\) with \(gcd (a_1, ... , a_{d+1}) = 1\) is called a regular system of weights, if the characteristic function \(\chi _{({\bold a},h)}(T) = (T^{h-a_1} - 1)...(T^{h-a_{d+1}} - 1) / ( T^{a_1}-1)...(T^{a_{d+1}} - 1) \) is a polynomial function (after Kyoji Saito, 1986). We show the following:

Theorem. For a pair of numbers \({\bold a} = (a_1, ... , a_{d+1})\), there is a number VF(\(\bold a\)) such that a regular system of weights \(({\bold a};h)\) with \(h > VF({{\bold a}})\) gives a weight system for a quasi-homogeneous complex analytic isolated singularity.

SUMMARY: We have developed a new method for constructing K3 surfaces. We constructed such a K3 surface \(X\) by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points. Our construction has \(19\) complex dimensional degrees of freedom. By the argument based on the concrete computation of the period map, we investigate which points in the period domain correspond to K3 surfaces obtained by such construction.

SUMMARY: In this talk, we study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem.

SUMMARY: Three different generalizations will be given for Nishino’s rigidity theorem asserting the triviality of Stein families of \(\mathbb {C}\) over the polydisc, in connection to generalizations of Hartogs’s theorem on the analyticity criterion for continuous functions.

SUMMARY: In this talk, we announce two mutually independent results on the family of Fock–Bargmann–Hartogs domains. Let \(D_1\) and \(D_2\) be two Fock–Bargmann–Hartogs domains in \(\mathbb {C}^{N_1}\) and \(\mathbb {C}^{N_2}\), respectively. In Theorem 1, we give a complete description of an arbitrarily given proper holomorphic mapping between \(D_1\) and \(D_2\) in the case where \(N_1 = N_2\). And, in Theorem 2, we determine the structure of \({\mathop {\rm Aut}}(D_1\times D_2)\) using the data of \({\mathop {\rm Aut}}(D_1)\) and \({\mathop {\rm Aut}}(D_2)\) for arbitrary \(N_1\) and \(N_2\).

SUMMARY: We introduce a new class of domains \(D_{n,m}(\mu ,p)\), called FBH-type domains, in \(\mathbb {C}^n\times \mathbb {C}^m\), where \(0<\mu \in \mathbb {R}\) and \(p\in \mathbb {N}\). In the special case of \(p=1\), these are just the Fock–Bargmann–Hartogs domains \(D_{n,m}(\mu )\) introduced by Yamamori. In this talk we give a complete description of a given proper holomorphic mapping between two equidimensional FBH-type domains. In particular, we prove that the holomorphic automorphism group of any FBH-type domain \(D_{n,m}(\mu ,p)\) with \(p\neq 1\) is a Lie group isomorphic to \(U(n)\times U(m)\). Hence the structure of \({\mathop {\rm Aut}}(D_{n,m}(\mu ,p))\) with \(p\neq 1\) is essentially different from that of \({\mathop {\rm Aut}}(D_{n,m}(\mu ))\).