Pub Date : 2024-05-15DOI: 10.4310/pamq.2024.v20.n3.a7
A. Garsia, G. Ganzberger
The Fibonacci polynomials ${lbrace F_n (x) rbrace}_{n geq 0}$ have been studied in multiple ways, [$href{https://www.imsc.res.in/~viennot/wa_files/viennotop1983-ocr.pdf}{1}$,$href{https://www.fq.math.ca/Scanned/11-3/hoggatt1.pdf}{6}$,$href{https://www.fq.math.ca/Scanned/12-2/hoggatt1.pdf}{7}$,$href{https://www.rivmat.unipr.it/fulltext/1995-4/1995-4-15.pdf}{9}$].In this paper we study them by means of the theory of heaps of Viennot [11, 12]. In this setting our polynomials form a basis ${lbrace P_n (x) rbrace}_{n geq 0}$ with $P_n (x)$ monic of degree $n$. This given, we are forced to set $P_n (x) = F_{n+1} (x)$. The heaps setting extends the Flajolet view$href{https://doi.org/10.1016/0012-365X(80)90050-3}{[4]}$ of the classical theory of orthogonal polynomials given by a three term recursion [3, 10]. Thus with heaps most of the identities for the polynomials $P_n (x)$’s can be derived by combinatorial arguments. Using the present setting we derive a variety of new identities. We must mention that the theory of heaps is presented here without restrictions. This is much more than needed to deal with the Fibonacci polynomials. We do this to convey a flavor of the power of heaps. In $href{https://link.springer.com/book/10.1007/978-3-030-58373-6 }{[5]}$ there is a chapter dedicated to heaps where most of its contents are dedicated to applications of the theory. In this paper we improve upon the developments in $href{https://link.springer.com/book/10.1007/978-3-030-58373-6 }{[5]}$ by adding details that were omitted there.
{"title":"Fibonacci polynomials","authors":"A. Garsia, G. Ganzberger","doi":"10.4310/pamq.2024.v20.n3.a7","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a7","url":null,"abstract":"The Fibonacci polynomials ${lbrace F_n (x) rbrace}_{n geq 0}$ have been studied in multiple ways, [$href{https://www.imsc.res.in/~viennot/wa_files/viennotop1983-ocr.pdf}{1}$,$href{https://www.fq.math.ca/Scanned/11-3/hoggatt1.pdf}{6}$,$href{https://www.fq.math.ca/Scanned/12-2/hoggatt1.pdf}{7}$,$href{https://www.rivmat.unipr.it/fulltext/1995-4/1995-4-15.pdf}{9}$].In this paper we study them by means of the theory of heaps of Viennot [11, 12]. In this setting our polynomials form a basis ${lbrace P_n (x) rbrace}_{n geq 0}$ with $P_n (x)$ monic of degree $n$. This given, we are forced to set $P_n (x) = F_{n+1} (x)$. The heaps setting extends the Flajolet view$href{https://doi.org/10.1016/0012-365X(80)90050-3}{[4]}$ of the classical theory of orthogonal polynomials given by a three term recursion [3, 10]. Thus with heaps most of the identities for the polynomials $P_n (x)$’s can be derived by combinatorial arguments. Using the present setting we derive a variety of new identities. We must mention that the theory of heaps is presented here without restrictions. This is much more than needed to deal with the Fibonacci polynomials. We do this to convey a flavor of the power of heaps. In $href{https://link.springer.com/book/10.1007/978-3-030-58373-6 }{[5]}$ there is a chapter dedicated to heaps where most of its contents are dedicated to applications of the theory. In this paper we improve upon the developments in $href{https://link.springer.com/book/10.1007/978-3-030-58373-6 }{[5]}$ by adding details that were omitted there.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-15DOI: 10.4310/pamq.2024.v20.n3.a4
Kevin Coulembier, Pavel Etingof, Victor Ostrik
Let $G$ be a group and $V$ a finite dimensional representation of $G$ over an algebraically closed field $k$ of characteristic $p gt 0$. Let $d_n (V)$ be the number of indecomposable summands of $V^{oplus n}$ of nonzero dimension $mod p$. It is easy to see that there exists a limit $delta (V) := lim_{n to infty} d_n(V)^{1/n}$, which is positive (and $geq 1$) $operatorname{iff}$ $V$ has an indecomposable summand of nonzero dimension $mod p$. We show that in this case the number[c(V ) := underset{n to infty}{lim inf} frac{d_n(V)}{delta(V)^n}$ in [0, 1]]is strictly positive and[log(c(V)^{-1}) = O(delta(V)^2),]and moreover this holds for any symmetric tensor category over $k$of moderate growth. Furthermore, we conjecture that in fact[log(c(V)^{-1}) = O(delta(V))](which would be sharp), and prove this for $p = 2, 3$; in particular, for $p = 2$ we show that $c(V) geq 3^{frac{4}{3} delta (V)+1}$. The proofs are based on the characteristic $p$ version of Deligne’s theorem for symmetric tensor categories obtained in $href{ https://dx.doi.org/10.4007/annals.2023.197.3.5}{[textrm{CEO}}]$. We also conjecture a classification of semisimple symmetric tensor categories of moderate growth which is interesting in its own right and implies the above conjecture for all $p$, and illustrate this conjecture by describing the semisimplification of the modular representation category of a cyclic $p$-group. Finally, we study the asymptotic behavior of the decomposition of $V^{oplus n}$ in characteristic zero using Deligne’s theorem and the Macdonald–Mehta–Opdam identity.
{"title":"Asymptotic properties of tensor powers in symmetric tensor categories","authors":"Kevin Coulembier, Pavel Etingof, Victor Ostrik","doi":"10.4310/pamq.2024.v20.n3.a4","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a4","url":null,"abstract":"Let $G$ be a group and $V$ a finite dimensional representation of $G$ over an algebraically closed field $k$ of characteristic $p gt 0$. Let $d_n (V)$ be the number of indecomposable summands of $V^{oplus n}$ of nonzero dimension $mod p$. It is easy to see that there exists a limit $delta (V) := lim_{n to infty} d_n(V)^{1/n}$, which is positive (and $geq 1$) $operatorname{iff}$ $V$ has an indecomposable summand of nonzero dimension $mod p$. We show that in this case the number[c(V ) := underset{n to infty}{lim inf} frac{d_n(V)}{delta(V)^n}$ in [0, 1]]is strictly positive and[log(c(V)^{-1}) = O(delta(V)^2),]and moreover this holds for any symmetric tensor category over $k$of moderate growth. Furthermore, we conjecture that in fact[log(c(V)^{-1}) = O(delta(V))](which would be sharp), and prove this for $p = 2, 3$; in particular, for $p = 2$ we show that $c(V) geq 3^{frac{4}{3} delta (V)+1}$. The proofs are based on the characteristic $p$ version of Deligne’s theorem for symmetric tensor categories obtained in $href{ https://dx.doi.org/10.4007/annals.2023.197.3.5}{[textrm{CEO}}]$. We also conjecture a classification of semisimple symmetric tensor categories of moderate growth which is interesting in its own right and implies the above conjecture for all $p$, and illustrate this conjecture by describing the semisimplification of the modular representation category of a cyclic $p$-group. Finally, we study the asymptotic behavior of the decomposition of $V^{oplus n}$ in characteristic zero using Deligne’s theorem and the Macdonald–Mehta–Opdam identity.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-15DOI: 10.4310/pamq.2024.v20.n3.a5
Alberto De Sole, Victor G. Kac, Daniele Valeri
The theory of triples of Poisson brackets and related integrable systems, based on a classical $R$-matrix $R in mathrm{End}_mathbb{F}(mathfrak{g})$, where $mathfrak{g}$ is a finite dimensional associative algebra over a field F viewed as a Lie algebra, was developed by Oevel–Ragnisco and Li–Parmentier [$href{https://doi.org/10.1016/0378-4371(89)90398-1}{textrm{OR89}}$, $href{https://doi.org/10.1007/BF01228340}{textrm{LP89}}$]. In the present paper we develop an “affine” analogue of this theory by introducing the notion of a continuous Poisson vertex algebra and constructing triples of Poisson $lambda$-brackets. We introduce the corresponding Adler type identities and apply them to integrability of hierarchies of Hamiltonian PDEs.
{"title":"Adler–Oevel-Ragnisco type operators and Poisson vertex algebras","authors":"Alberto De Sole, Victor G. Kac, Daniele Valeri","doi":"10.4310/pamq.2024.v20.n3.a5","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a5","url":null,"abstract":"The theory of triples of Poisson brackets and related integrable systems, based on a classical $R$-matrix $R in mathrm{End}_mathbb{F}(mathfrak{g})$, where $mathfrak{g}$ is a finite dimensional associative algebra over a field F viewed as a Lie algebra, was developed by Oevel–Ragnisco and Li–Parmentier [$href{https://doi.org/10.1016/0378-4371(89)90398-1}{textrm{OR89}}$, $href{https://doi.org/10.1007/BF01228340}{textrm{LP89}}$]. In the present paper we develop an “affine” analogue of this theory by introducing the notion of a continuous Poisson vertex algebra and constructing triples of Poisson $lambda$-brackets. We introduce the corresponding Adler type identities and apply them to integrability of hierarchies of Hamiltonian PDEs.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-15DOI: 10.4310/pamq.2024.v20.n3.a6
Giorgia Fortuna, Davide Lombardo, Andrea Maffei, Valerio Melani
In their study of spherical representations of an affine Lie algebra at the critical level and of unramified opers, Frenkel and Gaitsgory introduced what they called the Weyl module $mathbb{V}^lambda$ corresponding to a dominant weight $lambda$. This object plays an important role in the theory. In $href{ https://doi.org/10.1007/s00220-022-04430-w}{[4]}$, we introduced a possible analogue $mathbb{V}^{lambda,mu}_{2}$ of the Weyl module in the setting of opers with two singular points, and in the case of $mathfrak{sl}(2)$ we proved that it has the ‘correct’ endomorphism ring. In this paper, we compute the semi-infinite cohomology of $mathbb{V}^{lambda,mu}_{2}$ and we show that it does not share some of the properties of the semi-infinite cohomology of the Weyl module of Frenkel and Gaitsgory. For this reason, we introduce a new module $tilde{mathbb{V}}^{lambda,mu}_{2}$ which, in the case of $mathfrak{sl}(2)$, enjoys all the expected properties of a Weyl module.
{"title":"The semi-infinite cohomology of Weyl modules with two singular points","authors":"Giorgia Fortuna, Davide Lombardo, Andrea Maffei, Valerio Melani","doi":"10.4310/pamq.2024.v20.n3.a6","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n3.a6","url":null,"abstract":"In their study of spherical representations of an affine Lie algebra at the critical level and of unramified opers, Frenkel and Gaitsgory introduced what they called the <i>Weyl module</i> $mathbb{V}^lambda$ corresponding to a dominant weight $lambda$. This object plays an important role in the theory. In $href{ https://doi.org/10.1007/s00220-022-04430-w}{[4]}$, we introduced a possible analogue $mathbb{V}^{lambda,mu}_{2}$ of the Weyl module in the setting of opers with two singular points, and in the case of $mathfrak{sl}(2)$ we proved that it has the ‘correct’ endomorphism ring. In this paper, we compute the semi-infinite cohomology of $mathbb{V}^{lambda,mu}_{2}$ and we show that it does not share some of the properties of the semi-infinite cohomology of the Weyl module of Frenkel and Gaitsgory. For this reason, we introduce a new module $tilde{mathbb{V}}^{lambda,mu}_{2}$ which, in the case of $mathfrak{sl}(2)$, enjoys all the expected properties of a Weyl module.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-03DOI: 10.4310/pamq.2024.v20.n2.a3
Alexander Grigor’yan, Yong Lin, Shing-Tung Yau
We define the notions of Reidemeister torsion and analytic torsion for directed graphs by means of the path homology theory introduced by the authors in [ $href{https://arxiv.org/abs/1207.2834}{7}$, $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3324763}{8}$, $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3431683}{9}$, $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3845076}{11}$]. We prove the identity of the two notions of torsions as well as obtain formulas for torsions of Cartesian products and joins of digraphs.
{"title":"Analytic and Reidemeister torsions of digraphs and path complexes","authors":"Alexander Grigor’yan, Yong Lin, Shing-Tung Yau","doi":"10.4310/pamq.2024.v20.n2.a3","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n2.a3","url":null,"abstract":"We define the notions of Reidemeister torsion and analytic torsion for directed graphs by means of the path homology theory introduced by the authors in [ $href{https://arxiv.org/abs/1207.2834}{7}$, $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3324763}{8}$, $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3431683}{9}$, $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=3845076}{11}$]. We prove the identity of the two notions of torsions as well as obtain formulas for torsions of Cartesian products and joins of digraphs.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-03DOI: 10.4310/pamq.2024.v20.n2.a9
Mark Gross, Tyler L. Kelly, Ran J. Tessler
We show that a generating function for open $r$-spin enumerative invariants produces a universal unfolding of the polynomial $x^r$. Further, the coordinates parametrizing this universal unfolding are flat coordinates on the Frobenius manifold associated to the Landau–Ginzburg model $(mathbb{C}, x^r)$ via Saito–Givental theory. This result provides evidence for the same phenomenon to occur in higher dimension, proven in the sequel $href{https://arxiv.org/abs/2203.02435}{[textrm{GKT}22]}$.
{"title":"Mirror symmetry for open $r$-spin invariants","authors":"Mark Gross, Tyler L. Kelly, Ran J. Tessler","doi":"10.4310/pamq.2024.v20.n2.a9","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n2.a9","url":null,"abstract":"We show that a generating function for open $r$-spin enumerative invariants produces a universal unfolding of the polynomial $x^r$. Further, the coordinates parametrizing this universal unfolding are flat coordinates on the Frobenius manifold associated to the Landau–Ginzburg model $(mathbb{C}, x^r)$ via Saito–Givental theory. This result provides evidence for the same phenomenon to occur in higher dimension, proven in the sequel $href{https://arxiv.org/abs/2203.02435}{[textrm{GKT}22]}$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-03DOI: 10.4310/pamq.2024.v20.n2.a2
Ingmar Saberi, Brian R. Williams
We study a functor from two-step nilpotent super Lie algebras to sheaves of commutative differential graded algebras on the site of smooth $d$-manifolds, where $d$ is the dimension of the even subalgebra. The functor generalizes the pure spinor superfield formalism as studied in the physics literature. We prove that the functor commutes with deformations of the super Lie algebra by a Maurer–Cartan element, and apply the result to compute twists of various free supergravity theories and supersymmetric field theories of physical interest. Our results show that, just as the component fields of supersymmetric multiplets are the vector bundles associated to the equivariant Koszul homology of the variety of square-zero elements in the supersymmetry algebra, the component fields of the holomorphic twists of the corresponding multiplets are the holomorphic vector bundles associated to the equivariant Koszul homology of square-zero elements in the twisted supersymmetry algebra. The BRST or BV differentials of the free multiplet are induced by the brackets of the corresponding super Lie algebra in each case. We make this precise in a variety of examples; applications include rigorous computations of the minimal twists of eleven-dimensional and type IIB supergravity, in the free perturbative limit. The latter result proves a conjecture by Costello and Li, relating the IIB multiplet directly to a presymplectic BV version of minimal BCOV theory.
{"title":"Twisting pure spinor superfields, with applications to supergravity","authors":"Ingmar Saberi, Brian R. Williams","doi":"10.4310/pamq.2024.v20.n2.a2","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n2.a2","url":null,"abstract":"We study a functor from two-step nilpotent super Lie algebras to sheaves of commutative differential graded algebras on the site of smooth $d$-manifolds, where $d$ is the dimension of the even subalgebra. The functor generalizes the pure spinor superfield formalism as studied in the physics literature. We prove that the functor commutes with deformations of the super Lie algebra by a Maurer–Cartan element, and apply the result to compute twists of various free supergravity theories and supersymmetric field theories of physical interest. Our results show that, just as the component fields of supersymmetric multiplets are the vector bundles associated to the equivariant Koszul homology of the variety of square-zero elements in the supersymmetry algebra, the component fields of the holomorphic twists of the corresponding multiplets are the holomorphic vector bundles associated to the equivariant Koszul homology of square-zero elements in the twisted supersymmetry algebra. The BRST or BV differentials of the free multiplet are induced by the brackets of the corresponding super Lie algebra in each case. We make this precise in a variety of examples; applications include rigorous computations of the minimal twists of eleven-dimensional and type IIB supergravity, in the free perturbative limit. The latter result proves a conjecture by Costello and Li, relating the IIB multiplet directly to a presymplectic BV version of minimal BCOV theory.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-03DOI: 10.4310/pamq.2024.v20.n2.a4
Dawei Shen
In 1993, the global stability of Minkowski spacetime has been proven in the celebrated work of Christodoulou and Klainerman $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1316662}{[5]}$ in a maximal foliation. In 2003, Klainerman and Nicolò $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1946854}{[14]}$ gave a second proof of the stability of Minkowski in the case of the exterior of an outgoing null cone. In this paper, we give a new proof of $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1946854}{[14]}$. Compared to $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1946854}{[14]}$, we reduce the number of derivatives needed in the proof, simplify the treatment of the last slice, and provide a unified treatment of the decay of initial data. Also, concerning the treatment of curvature estimates, we replace the vectorfield method used in $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1946854}{[14]}$ by the $r^p$-weighted estimates of Dafermos and Rodnianski $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=2730803}{[7]}$.
{"title":"Stability of Minkowski spacetime in exterior regions","authors":"Dawei Shen","doi":"10.4310/pamq.2024.v20.n2.a4","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n2.a4","url":null,"abstract":"In 1993, the global stability of Minkowski spacetime has been proven in the celebrated work of Christodoulou and Klainerman $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1316662}{[5]}$ in a maximal foliation. In 2003, Klainerman and Nicolò $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1946854}{[14]}$ gave a second proof of the stability of Minkowski in the case of the exterior of an outgoing null cone. In this paper, we give a new proof of $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1946854}{[14]}$. Compared to $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1946854}{[14]}$, we reduce the number of derivatives needed in the proof, simplify the treatment of the last slice, and provide a unified treatment of the decay of initial data. Also, concerning the treatment of curvature estimates, we replace the vectorfield method used in $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1946854}{[14]}$ by the $r^p$-weighted estimates of Dafermos and Rodnianski $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=2730803}{[7]}$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-03DOI: 10.4310/pamq.2024.v20.n2.a1
Sven Hirsch, Pengzi Miao, Luen-Fai Tam
We derive local and global monotonic quantities associated to $p$-harmonic functions on manifolds with nonnegative scalar curvature. As applications, we obtain inequalities relating the mass of asymptotically flat $3$-manifolds, the $p$-capacity and the Willmore functional of the boundary. As $p to 1$, one of the results retrieves a classic relation that the ADM mass dominates the Hawking mass if the surface is area outer-minimizing.
我们推导了具有非负标量曲率的流形上与 $p$ 谐函数相关的局部和全局单调量。作为应用，我们得到了与渐近平坦的$3$流形的质量、$p$容量和边界的威尔摩尔函数相关的不等式。当 $p to 1$ 时，其中一个结果检索到一个经典关系，即如果曲面是面积外最小的，则 ADM 质量支配霍金质量。
{"title":"Monotone quantities of $p$-harmonic functions and their applications","authors":"Sven Hirsch, Pengzi Miao, Luen-Fai Tam","doi":"10.4310/pamq.2024.v20.n2.a1","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n2.a1","url":null,"abstract":"We derive local and global monotonic quantities associated to $p$-harmonic functions on manifolds with nonnegative scalar curvature. As applications, we obtain inequalities relating the mass of asymptotically flat $3$-manifolds, the $p$-capacity and the Willmore functional of the boundary. As $p to 1$, one of the results retrieves a classic relation that the ADM mass dominates the Hawking mass if the surface is area outer-minimizing.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-03DOI: 10.4310/pamq.2024.v20.n2.a6
Akito Futaki
We consider weighted solitons on Fano manifolds which include Kähler–Ricci solitons, Mabuchi solitons and base metrics inducing Calabi–Yau cone metrics outside the zero sections of the canonical line bundles (Sasaki–Einstein metrics on the associated $U(1)$-bundles). In this paper, we give a condition for a weighted soliton on a Fano manifold $M_0$ to extend to weighted solitons on small deformations $M_t$ of the Fano manifold $M_0$. More precisely, we show that all the members $M_t$ of the Kuranishi family of a Fano manifold $M_0$ with a weighted soliton have weighted solitons if and only if the dimensions of $T$-equivariant automorphism groups of $M_t$ are equal to that of $M_0$, and also if and only if the $T$-equivariant automorphism groups of $M_t$ are all isomorphic to that of $M_0$, where the weight functions are defined on the moment polytope of the Hamiltonian $T$-action. This generalizes a result of Cao–Sun–Yau–Zhang for Kähler–Einstein metrics.
{"title":"Deformations of Fano manifolds with weighted solitons","authors":"Akito Futaki","doi":"10.4310/pamq.2024.v20.n2.a6","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n2.a6","url":null,"abstract":"We consider weighted solitons on Fano manifolds which include Kähler–Ricci solitons, Mabuchi solitons and base metrics inducing Calabi–Yau cone metrics outside the zero sections of the canonical line bundles (Sasaki–Einstein metrics on the associated $U(1)$-bundles). In this paper, we give a condition for a weighted soliton on a Fano manifold $M_0$ to extend to weighted solitons on small deformations $M_t$ of the Fano manifold $M_0$. More precisely, we show that all the members $M_t$ of the Kuranishi family of a Fano manifold $M_0$ with a weighted soliton have weighted solitons if and only if the dimensions of $T$-equivariant automorphism groups of $M_t$ are equal to that of $M_0$, and also if and only if the $T$-equivariant automorphism groups of $M_t$ are all isomorphic to that of $M_0$, where the weight functions are defined on the moment polytope of the Hamiltonian $T$-action. This generalizes a result of Cao–Sun–Yau–Zhang for Kähler–Einstein metrics.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：