Pub Date : 2020-10-07DOI: 10.1512/iumj.2023.72.9211
R. Cristoferi, G. Gravina
We provide the integral representation formula for the relaxation in $BV(o; R^M)$ with respect to strong convergence in $L^1(o; R^M)$ of a functional with a boundary contact energy term. This characterization is valid for a large class of surface energy densities, and for domains satisfying mild regularity assumptions. Motivated by some classical examples where lower semicontinuity fails, we analyze the extent to which the geometry of the set enters the relaxation procedure.
{"title":"On the relaxation of functionals with contact terms on non-smooth domains","authors":"R. Cristoferi, G. Gravina","doi":"10.1512/iumj.2023.72.9211","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9211","url":null,"abstract":"We provide the integral representation formula for the relaxation in $BV(o; R^M)$ with respect to strong convergence in $L^1(o; R^M)$ of a functional with a boundary contact energy term. This characterization is valid for a large class of surface energy densities, and for domains satisfying mild regularity assumptions. Motivated by some classical examples where lower semicontinuity fails, we analyze the extent to which the geometry of the set enters the relaxation procedure.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41829456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-04DOI: 10.1512/iumj.2022.71.9042
M. Winkler
The parabolic-elliptic cross-diffusion system [ left{ begin{array}{l} u_t = Delta u - nabla cdot Big(uf(|nabla v|^2) nabla v Big), [1mm] 0 = Delta v - mu + u, qquad int_Omega v=0, qquad mu:=frac{1}{|Omega|} int_Omega u dx, end{array} right. ] is considered along with homogeneous Neumann-type boundary conditions in a smoothly bounded domain $Omegasubset R^n$, $nge 1$, where $f$ generalizes the prototype given by [ f(xi) = (1+xi)^{-alpha}, qquad xige 0, qquad mbox{for all } xige 0, ] with $alphain R$. In this framework, the main results assert that if $nge 2$, $Omega$ is a ball and [ alphafrac{n-2}{2(n-1)}$, then any explosion is ruled out in the sense that for arbitrary nonnegative and continuous initial data, a global bounded classical solution exists.
抛物型椭圆交叉扩散系统[left{begin{array}｛l｝u_t=Delta u-nablacdotBig（uf（|nablav|^2）nablaVBig），[1mm]0=Delta v-mu+u，qquadint_Omega v=0，qqaudmu:=frac｛1｝与光滑有界域$Omegasubet R^n$，$nge 1$中的齐次Neumann型边界条件一起考虑，其中$f$推广了由[f（neneneba xi）=（1+nenenebb xi）^｛-alpha｝，qquadnenenebc xi ge 0，qqaudmbox｛for all｝nenenebd xi ge0，]给出的原型，其中$alpha在R$中。在这个框架中，主要结果断言，如果$nge2$，$Omega$是一个球，并且[alphafrac｛n-2｝｛2（n-1）｝$，则在任意非负和连续初始数据存在全局有界经典解的意义上，排除了任何爆炸。
{"title":"A critical blow-up exponent for flux limiation in a Keller-Segel system","authors":"M. Winkler","doi":"10.1512/iumj.2022.71.9042","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9042","url":null,"abstract":"The parabolic-elliptic cross-diffusion system [ \u0000left{ begin{array}{l} \u0000u_t = Delta u - nabla cdot Big(uf(|nabla v|^2) nabla v Big), [1mm] \u00000 = Delta v - mu + u, \u0000qquad int_Omega v=0, \u0000qquad \u0000mu:=frac{1}{|Omega|} int_Omega u dx, \u0000end{array} right. ] is considered along with homogeneous Neumann-type boundary conditions in a smoothly bounded domain $Omegasubset R^n$, $nge 1$, where $f$ generalizes the prototype given by [ \u0000f(xi) = (1+xi)^{-alpha}, \u0000qquad xige 0, \u0000qquad mbox{for all } xige 0, ] with $alphain R$. \u0000In this framework, the main results assert that if $nge 2$, $Omega$ is a ball and [ \u0000alpha<frac{n-2}{2(n-1)}, ] then throughout a considerably large set of radially symmetric initial data, an associated initial value problem admits solutions blowing up in finite time with respect to the $L^infty$ norm of their first components. \u0000This is complemented by a second statement which ensures that in general and not necessarily symmetric settings, if either $n=1$ and $alphain R$ is arbitrary, or $nge 2$ and $alpha>frac{n-2}{2(n-1)}$, then any explosion is ruled out in the sense that for arbitrary nonnegative and continuous initial data, a global bounded classical solution exists.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47936083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-22DOI: 10.1512/iumj.2023.72.9279
Adem Limani, A. Nicolau
We study analogues of well-known relationships between Muckenhoupt weights and $BMO$ in the setting of Bekolle-Bonami weights. For Bekolle-Bonami weights of bounded hyperbolic oscillation, we provide distance formulas of Garnett and Jones-type, in the context of $BMO$ on the unit disc and hyperbolic Lipschitz functions. This leads to a characterization of all weights in this class, for which any power of the weight is a Bekolle-Bonami weight, which in particular reveals an intimate connection between Bekolle-Bonami weights and Bloch functions. On the open problem of characterizing the closure of bounded analytic functions in the Bloch space, we provide a counter-example to a related recent conjecture. This shed light into the difficulty of preserving harmonicity in approximation problems in norms equivalent to the Bloch norm. Finally, we apply our results to study certain spectral properties of Cesaro operators.
{"title":"Bloch functions and Bekolle-Bonami weights","authors":"Adem Limani, A. Nicolau","doi":"10.1512/iumj.2023.72.9279","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9279","url":null,"abstract":"We study analogues of well-known relationships between Muckenhoupt weights and $BMO$ in the setting of Bekolle-Bonami weights. For Bekolle-Bonami weights of bounded hyperbolic oscillation, we provide distance formulas of Garnett and Jones-type, in the context of $BMO$ on the unit disc and hyperbolic Lipschitz functions. This leads to a characterization of all weights in this class, for which any power of the weight is a Bekolle-Bonami weight, which in particular reveals an intimate connection between Bekolle-Bonami weights and Bloch functions. On the open problem of characterizing the closure of bounded analytic functions in the Bloch space, we provide a counter-example to a related recent conjecture. This shed light into the difficulty of preserving harmonicity in approximation problems in norms equivalent to the Bloch norm. Finally, we apply our results to study certain spectral properties of Cesaro operators.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47040347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-14DOI: 10.1512/iumj.2023.72.9485
U. Franz, A. Kula, J. Lindsay, Michael Skeide
We provide a Hunt type formula for the infinitesimal generators of Levy process on the quantum groups $SU_q(N)$ and $U_q(N)$. In particular, we obtain a decomposition of such generators into a gaussian part and a `jump type' part determined by a linear functional that resembles the functional induced by the Levy measure. The jump part on $SU_q(N)$ decomposes further into parts that live on the quantum subgroups $SU_q(n)$, $nle N$. Like in the classical Hunt formula for locally compact Lie groups, the ingredients become unique once a certain projection is chosen. There are analogous result for $U_q(N)$.
{"title":"Hunt's formula for SU_q(N) and U_q(N)","authors":"U. Franz, A. Kula, J. Lindsay, Michael Skeide","doi":"10.1512/iumj.2023.72.9485","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9485","url":null,"abstract":"We provide a Hunt type formula for the infinitesimal generators of Levy process on the quantum groups $SU_q(N)$ and $U_q(N)$. In particular, we obtain a decomposition of such generators into a gaussian part and a `jump type' part determined by a linear functional that resembles the functional induced by the Levy measure. The jump part on $SU_q(N)$ decomposes further into parts that live on the quantum subgroups $SU_q(n)$, $nle N$. Like in the classical Hunt formula for locally compact Lie groups, the ingredients become unique once a certain projection is chosen. There are analogous result for $U_q(N)$.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48677569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-04DOI: 10.1512/iumj.2022.71.9164
Grégory Faye, Matt Holzer, A. Scheel, L. Siemer
We study the invasion of an unstable state by a propagating front in a peculiar but generic situation where the invasion process exhibits a remnant instability. Here, remnant instability refers to the fact that the spatially constant invaded state is linearly unstable in any exponentially weighted space in a frame moving with the linear invasion speed. Our main result is the nonlinear asymptotic stability of the selected invasion front for a prototypical model coupling spatio-temporal oscillations and monotone dynamics. We establish stability through a decomposition of the perturbation into two pieces: one that is bounded in the weighted space and a second that is unbounded in the weighted space but which converges uniformly to zero in the unweighted space at an exponential rate. Interestingly, long-time numerical simulations reveal an apparent instability in some cases. We exhibit how this instability is caused by round-off errors that introduce linear resonant coupling of otherwise non-resonant linear modes, and we determine the accelerated invasion speed.
{"title":"Invasion into remnant instability: a case study of front dynamics","authors":"Grégory Faye, Matt Holzer, A. Scheel, L. Siemer","doi":"10.1512/iumj.2022.71.9164","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9164","url":null,"abstract":"We study the invasion of an unstable state by a propagating front in a peculiar but generic situation where the invasion process exhibits a remnant instability. Here, remnant instability refers to the fact that the spatially constant invaded state is linearly unstable in any exponentially weighted space in a frame moving with the linear invasion speed. Our main result is the nonlinear asymptotic stability of the selected invasion front for a prototypical model coupling spatio-temporal oscillations and monotone dynamics. We establish stability through a decomposition of the perturbation into two pieces: one that is bounded in the weighted space and a second that is unbounded in the weighted space but which converges uniformly to zero in the unweighted space at an exponential rate. Interestingly, long-time numerical simulations reveal an apparent instability in some cases. We exhibit how this instability is caused by round-off errors that introduce linear resonant coupling of otherwise non-resonant linear modes, and we determine the accelerated invasion speed.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43792958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-03DOI: 10.1512/iumj.2022.71.9163
D. Shlyakhtenko, Terence Tao. With an appendix by David Jekel
The extension $k mapsto mu^{boxplus k}$ of the concept of a free convolution power to the case of non-integer $k geq 1$ was introduced by Bercovici-Voiculescu and Nica-Speicher, and related to the minor process in random matrix theory. In this paper we give two proofs of the monotonicity of the free entropy and free Fisher information of the (normalized) free convolution power in this continuous setting, and also establish an intriguing variational description of this process.
{"title":"Fractional free convolution powers","authors":"D. Shlyakhtenko, Terence Tao. With an appendix by David Jekel","doi":"10.1512/iumj.2022.71.9163","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9163","url":null,"abstract":"The extension $k mapsto mu^{boxplus k}$ of the concept of a free convolution power to the case of non-integer $k geq 1$ was introduced by Bercovici-Voiculescu and Nica-Speicher, and related to the minor process in random matrix theory. In this paper we give two proofs of the monotonicity of the free entropy and free Fisher information of the (normalized) free convolution power in this continuous setting, and also establish an intriguing variational description of this process.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45148558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-02DOI: 10.1512/iumj.2023.72.9333
N. Hamamuki, S. Kikkawa
We derive a lower spatially Lipschitz bound for viscosity solutions to fully nonlinear parabolic partial differential equations when the initial datum belongs to the H(cid:127)older space. The resulting estimate depends on the initial H(cid:127)older exponent and the growth rates of the equation with respect to the (cid:12)rst and second order derivative terms. Our estimate is applicable to equations which are possibly singular at the initial time. Moreover, it gives the optimal rate of the regularizing effect for solutions, which occurs for some uniformly parabolic equations and (cid:12)rst order Hamilton{Jacobi equations. In the proof of our lower estimate, we construct a subsolution and a supersolution by optimally rescaling the solution of the heat equation and then compare them with the solution. For linear equations, the lower spatially Lipschitz bound for solutions can be obtained in a different way if the fundamental solution satis(cid:12)es the Aronson estimate. Examples include the heat convection equation whose convection term has singularities.
{"title":"A lower spatially Lipschitz bound for solutions to fully nonlinear parabolic equations and its optimality","authors":"N. Hamamuki, S. Kikkawa","doi":"10.1512/iumj.2023.72.9333","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9333","url":null,"abstract":"We derive a lower spatially Lipschitz bound for viscosity solutions to fully nonlinear parabolic partial differential equations when the initial datum belongs to the H(cid:127)older space. The resulting estimate depends on the initial H(cid:127)older exponent and the growth rates of the equation with respect to the (cid:12)rst and second order derivative terms. Our estimate is applicable to equations which are possibly singular at the initial time. Moreover, it gives the optimal rate of the regularizing effect for solutions, which occurs for some uniformly parabolic equations and (cid:12)rst order Hamilton{Jacobi equations. In the proof of our lower estimate, we construct a subsolution and a supersolution by optimally rescaling the solution of the heat equation and then compare them with the solution. For linear equations, the lower spatially Lipschitz bound for solutions can be obtained in a different way if the fundamental solution satis(cid:12)es the Aronson estimate. Examples include the heat convection equation whose convection term has singularities.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66765696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-27DOI: 10.1512/iumj.2022.71.9386
Ben Hayes
The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free group factor is contained in a unique maximal amenable subalgebra. This conjecture is motivated by related results in Popa's deformation/rigidity theory and Peterson-Thom's results on L^{2}-Betti numbers. We present an approach to this conjecture in terms of so-called strong convergence of random matrices by formulating a conjecture which is a natural generalization of the Haagerup-Thorbjornsen theorem whose validity would imply the Peterson-Thom conjecture. This random matrix conjecture is related to recent work of Collins-Guionnet-Parraud.
{"title":"A random matrix approach to the Peterson-Thom conjecture","authors":"Ben Hayes","doi":"10.1512/iumj.2022.71.9386","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9386","url":null,"abstract":"The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free group factor is contained in a unique maximal amenable subalgebra. This conjecture is motivated by related results in Popa's deformation/rigidity theory and Peterson-Thom's results on L^{2}-Betti numbers. We present an approach to this conjecture in terms of so-called strong convergence of random matrices by formulating a conjecture which is a natural generalization of the Haagerup-Thorbjornsen theorem whose validity would imply the Peterson-Thom conjecture. This random matrix conjecture is related to recent work of Collins-Guionnet-Parraud.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45133211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-24DOI: 10.1512/iumj.2023.72.9408
Jason DeVito, F. Galaz‐García, M. Kerin
Under mild topological restrictions, this article establishes that a smooth, closed, simply connected manifold of dimension at most seven which can be decomposed as the union of two disk bundles must be rationally elliptic. In dimension five, such manifolds are classified up to diffeomorphism, while the same is true in dimension six when either the second Betti number vanishes or the third Betti number is non-trivial.
{"title":"Manifolds that admit a double disk-bundle decomposition","authors":"Jason DeVito, F. Galaz‐García, M. Kerin","doi":"10.1512/iumj.2023.72.9408","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9408","url":null,"abstract":"Under mild topological restrictions, this article establishes that a smooth, closed, simply connected manifold of dimension at most seven which can be decomposed as the union of two disk bundles must be rationally elliptic. In dimension five, such manifolds are classified up to diffeomorphism, while the same is true in dimension six when either the second Betti number vanishes or the third Betti number is non-trivial.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42277551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-16DOI: 10.1512/iumj.2023.72.9405
Seungjae Lee, Aeryeong Seo
Let $Sigma = mathbb B^n/Gamma$ be a compact complex hyperbolic space with torsion-free lattice $Gammasubset SU(n,1)$ and $Omega $ a quotient of $mathbb B^n timesmathbb B^n$ with respect to the diagonal action of $Gamma$ which is a holomorphic $mathbb B^n$-fiber bundle over $Sigma$. The goal of this article is to investigate the relation between symmetric differentials of $Sigma$ and the weighted $L^2$ holomorphic functions on the exhaustions $Omega_epsilon$ of $Omega$. If there exists a holomorphic function on $Omega_epsilon$ on some $epsilon$, then there exists a symmetric differential on $Sigma$. Using this property, we show that $Sigma$ always has a symmetric differential of degree $N$ for any $Ngeq n+1$. Moreover for each symmetric differential over $Sigma$, we construct a weighted $L^2$ holomorphic function on $Omega_{1over sqrt{n}}$.
设$Sigma=mathbb B^n/Gamma$是一个紧致复双曲空间，其无扭格$Gamma子集SU（n，1）$和$Omega$是$mathbb B ^ntimesmathbb B^n$关于$Gamma$的对角作用的商，$Gamma$是$Sigma$上的全纯$mathbbB^n$-纤维束。本文的目的是研究$Sigma$的对称微分与$Omega_epsilon$的穷举$Omega上的加权$L^2$全纯函数之间的关系。如果在$Omega_epsilon$上存在一个全纯函数，则在$Sigma$上存在对称微分。利用这个性质，我们证明了对于任何$Ngeqn+1$，$Sigma$总是具有次为$N$的对称微分。此外，对于$Sigma$上的每个对称微分，我们构造了$Omega_｛1oversqrt｛n｝｝$上的加权$L^2$全纯函数。
{"title":"Symmetric differentials and Jets extension of L^2 holomorphic functions","authors":"Seungjae Lee, Aeryeong Seo","doi":"10.1512/iumj.2023.72.9405","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9405","url":null,"abstract":"Let $Sigma = mathbb B^n/Gamma$ be a compact complex hyperbolic space with torsion-free lattice $Gammasubset SU(n,1)$ and $Omega $ a quotient of $mathbb B^n timesmathbb B^n$ with respect to the diagonal action of $Gamma$ which is a holomorphic $mathbb B^n$-fiber bundle over $Sigma$. The goal of this article is to investigate the relation between symmetric differentials of $Sigma$ and the weighted $L^2$ holomorphic functions on the exhaustions $Omega_epsilon$ of $Omega$. If there exists a holomorphic function on $Omega_epsilon$ on some $epsilon$, then there exists a symmetric differential on $Sigma$. Using this property, we show that $Sigma$ always has a symmetric differential of degree $N$ for any $Ngeq n+1$. Moreover for each symmetric differential over $Sigma$, we construct a weighted $L^2$ holomorphic function on $Omega_{1over sqrt{n}}$.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47407456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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