In $d$-dimensional space (over any field), given a set of lines, a joint is a point passed through by $d$ lines not all lying in some hyperplane. The joints problem asks to determine the maximum number of joints formed by $L$ lines, and it was one of the successes of the Guth--Katz polynomial method. We prove a new upper bound on the number of joints that matches, up to a $1+o(1)$ factor, the best known construction: place $k$ generic hyperplanes, and use their $(d-1)$-wise intersections to form $binom{k}{d-1}$ lines and their $d$-wise intersections to form $binom{k}{d}$ joints. Guth conjectured that this construction is optimal. Our technique builds on the work on Ruixiang Zhang proving the multijoints conjecture via an extension of the polynomial method. We set up a variational problem to control the high order of vanishing of a polynomial at each joint.
{"title":"Joints tightened","authors":"H. Yu, Yufei Zhao","doi":"10.1353/ajm.2023.0014","DOIUrl":"https://doi.org/10.1353/ajm.2023.0014","url":null,"abstract":"In $d$-dimensional space (over any field), given a set of lines, a joint is a point passed through by $d$ lines not all lying in some hyperplane. The joints problem asks to determine the maximum number of joints formed by $L$ lines, and it was one of the successes of the Guth--Katz polynomial method. We prove a new upper bound on the number of joints that matches, up to a $1+o(1)$ factor, the best known construction: place $k$ generic hyperplanes, and use their $(d-1)$-wise intersections to form $binom{k}{d-1}$ lines and their $d$-wise intersections to form $binom{k}{d}$ joints. Guth conjectured that this construction is optimal. Our technique builds on the work on Ruixiang Zhang proving the multijoints conjecture via an extension of the polynomial method. We set up a variational problem to control the high order of vanishing of a polynomial at each joint.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46016767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract:In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem hasdeserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Maric{s} in 2013. However, such result is valid only in dimension 3 and higher. In this paperwe first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3.With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3,recovering the aforementioned result by Maric{s}. The planar case turns out to be more intricate and the compactness argumentworks only under an additional assumption on the vortex set of the approximating solutions.
{"title":"Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime","authors":"J. Bellazzini, D. Ruiz","doi":"10.1353/ajm.2023.0002","DOIUrl":"https://doi.org/10.1353/ajm.2023.0002","url":null,"abstract":"Abstract:In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem hasdeserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Maric{s} in 2013. However, such result is valid only in dimension 3 and higher. In this paperwe first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3.With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3,recovering the aforementioned result by Maric{s}. The planar case turns out to be more intricate and the compactness argumentworks only under an additional assumption on the vortex set of the approximating solutions.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"20 1","pages":"109 - 149"},"PeriodicalIF":1.7,"publicationDate":"2019-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66915686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract:Though there has been extensive study on Hardy-Sobolev-Maz'ya inequalities on upper half spaces for first order derivatives, whether an analogous inequality for higher order derivatives holds has still remained open. By using, among other things, the Fourier analysis techniques on the hyperbolic space which is a noncompact complete Riemannian manifold, we establish the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces. Moreover, we derive sharp Poincar'e-Sobolev inequalities (namely, Sobolev inequalities with a substraction of a Hardy term) for the Paneitz operators on hyperbolic spaces which are of their independent interests and useful in establishing the sharp Hardy-Sobolev-Maz'ya inequalities. Our sharp Poincar'e-Sobolev inequalities for the Paneitz operators on hyperbolic spaces improve substantially those Sobolev inequalities in the literature. The proof of such Poincar'e-Sobolev inequalities relies on hard analysis of Green's functions estimates, Fourier analysis on hyperbolic spaces together with the Hardy-Littlewood-Sobolev inequality on the hyperbolic spaces. Finally, we show the sharp constant in the Hardy-Sobolev-Maz'ya inequality for the bi-Laplacian in the upper half space of dimension five coincides with the best Sobolev constant. This is an analogous result to that of the sharp constant in the first order Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half spaces.
{"title":"Paneitz operators on hyperbolic spaces and high order Hardy-Sobolev-Maz'ya inequalities on half spaces","authors":"Guozhen Lu, Qiaohua Yang","doi":"10.1353/ajm.2019.0047","DOIUrl":"https://doi.org/10.1353/ajm.2019.0047","url":null,"abstract":"Abstract:Though there has been extensive study on Hardy-Sobolev-Maz'ya inequalities on upper half spaces for first order derivatives, whether an analogous inequality for higher order derivatives holds has still remained open. By using, among other things, the Fourier analysis techniques on the hyperbolic space which is a noncompact complete Riemannian manifold, we establish the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces. Moreover, we derive sharp Poincar'e-Sobolev inequalities (namely, Sobolev inequalities with a substraction of a Hardy term) for the Paneitz operators on hyperbolic spaces which are of their independent interests and useful in establishing the sharp Hardy-Sobolev-Maz'ya inequalities. Our sharp Poincar'e-Sobolev inequalities for the Paneitz operators on hyperbolic spaces improve substantially those Sobolev inequalities in the literature. The proof of such Poincar'e-Sobolev inequalities relies on hard analysis of Green's functions estimates, Fourier analysis on hyperbolic spaces together with the Hardy-Littlewood-Sobolev inequality on the hyperbolic spaces. Finally, we show the sharp constant in the Hardy-Sobolev-Maz'ya inequality for the bi-Laplacian in the upper half space of dimension five coincides with the best Sobolev constant. This is an analogous result to that of the sharp constant in the first order Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half spaces.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"141 1","pages":"1777 - 1816"},"PeriodicalIF":1.7,"publicationDate":"2019-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2019.0047","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41880483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract:We adapt to an infinite dimensional ambient space E. R. Reifenberg's epiperimetric inequality and a quantitative version of D. Preiss' second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable $G$ chain in $ell_2$ is dense in its support, whenever the group $G$ of coefficients is so that ${|g|:gin G}$ is discrete.
{"title":"Partial regularity of almost minimizing rectifiable G chains in Hilbert space","authors":"Thierry de Pauw, Roger Züst","doi":"10.1353/ajm.2019.0044","DOIUrl":"https://doi.org/10.1353/ajm.2019.0044","url":null,"abstract":"Abstract:We adapt to an infinite dimensional ambient space E. R. Reifenberg's epiperimetric inequality and a quantitative version of D. Preiss' second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable $G$ chain in $ell_2$ is dense in its support, whenever the group $G$ of coefficients is so that ${|g|:gin G}$ is discrete.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"141 1","pages":"1591 - 1705"},"PeriodicalIF":1.7,"publicationDate":"2019-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2019.0044","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44953003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract:Using arithmetic jet spaces we attach perfectoid spaces to smooth schemes and we attach morphisms of perfectoidspaces to $delta$-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable $delta$-morphisms appearing in the theory such as the $delta$-charactersof elliptic curves and the $delta$-period maps on modular curves.
{"title":"Perfectoid spaces arising from arithmetic jet spaces","authors":"A. Buium, L. Miller","doi":"10.1353/ajm.2023.0006","DOIUrl":"https://doi.org/10.1353/ajm.2023.0006","url":null,"abstract":"Abstract:Using arithmetic jet spaces we attach perfectoid spaces to smooth schemes and we attach morphisms of perfectoidspaces to $delta$-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable $delta$-morphisms appearing in the theory such as the $delta$-charactersof elliptic curves and the $delta$-period maps on modular curves.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"145 1","pages":"287 - 334"},"PeriodicalIF":1.7,"publicationDate":"2019-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44330080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-15DOI: 10.1353/ajm.2023.a902953
F. Bonsante, Gabriele Mondello, Jean-Marc Schlenker
abstract:Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian $3$-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These ``minimizing'' maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps---but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in $M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a given homotopy class. When $(S,h)$ is fixed, smooth minimizing maps from $(S,h)$ are described by a simple holomorphic data on $S$: a complex self-adjoint Codazzi tensor of determinant $1$. The space of admissible data is smooth and naturally equipped with a complex structure, for which the monodromy map taking a data to the monodromy representation of the image is holomorphic. Minimizing maps are in this way reminiscent of shear-bend coordinates, with the complexification of $F$ analoguous to the complex length.
{"title":"Minimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold","authors":"F. Bonsante, Gabriele Mondello, Jean-Marc Schlenker","doi":"10.1353/ajm.2023.a902953","DOIUrl":"https://doi.org/10.1353/ajm.2023.a902953","url":null,"abstract":"abstract:Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian $3$-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These ``minimizing'' maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps---but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in $M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a given homotopy class. When $(S,h)$ is fixed, smooth minimizing maps from $(S,h)$ are described by a simple holomorphic data on $S$: a complex self-adjoint Codazzi tensor of determinant $1$. The space of admissible data is smooth and naturally equipped with a complex structure, for which the monodromy map taking a data to the monodromy representation of the image is holomorphic. Minimizing maps are in this way reminiscent of shear-bend coordinates, with the complexification of $F$ analoguous to the complex length.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"145 1","pages":"1049 - 995"},"PeriodicalIF":1.7,"publicationDate":"2019-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45403767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract:We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity.
{"title":"Optimal boundary regularity for fast diffusion equations in bounded domains","authors":"Tianling Jin, Jingang Xiong","doi":"10.1353/ajm.2023.0003","DOIUrl":"https://doi.org/10.1353/ajm.2023.0003","url":null,"abstract":"Abstract:We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"145 1","pages":"151 - 219"},"PeriodicalIF":1.7,"publicationDate":"2019-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45208377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
abstract:Extending ideas of Hedden-Ni, we show that the module structure on Khovanov homology detects split links. We also prove an analogue for untwisted Heegaard Floer homology of the branched double cover. Technical results proved along the way include two interpretations of the module structure on untwisted Heegaard Floer homology in terms of twisted Heegaard Floer homology and the fact that the module structure on the reduced Khovanov complex of a link is well defined up to quasi-isomorphism.
{"title":"Khovanov homology detects split links","authors":"Robert Lipshitz, Sucharit Sarkar","doi":"10.1353/ajm.2022.0043","DOIUrl":"https://doi.org/10.1353/ajm.2022.0043","url":null,"abstract":"abstract:Extending ideas of Hedden-Ni, we show that the module structure on Khovanov homology detects split links. We also prove an analogue for untwisted Heegaard Floer homology of the branched double cover. Technical results proved along the way include two interpretations of the module structure on untwisted Heegaard Floer homology in terms of twisted Heegaard Floer homology and the fact that the module structure on the reduced Khovanov complex of a link is well defined up to quasi-isomorphism.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"144 1","pages":"1745 - 1781"},"PeriodicalIF":1.7,"publicationDate":"2019-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46860615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
abstract:We study the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the result of Guth--Iosevich--Ou--Wang for the distance set in the plane to general Riemannian surfaces. Key new ingredients include a family of refined microlocal decoupling inequalities, which are related to the work of Beltran--Hickman--Sogge on Wolff-type inequalities, and an analog of Orponen's radial projection lemma which has proved quite useful in recent work on distance sets.
{"title":"Microlocal decoupling inequalities and the distance problem on Riemannian manifolds","authors":"A. Iosevich, Bochen Liu, Yakun Xi","doi":"10.1353/ajm.2022.0039","DOIUrl":"https://doi.org/10.1353/ajm.2022.0039","url":null,"abstract":"abstract:We study the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the result of Guth--Iosevich--Ou--Wang for the distance set in the plane to general Riemannian surfaces. Key new ingredients include a family of refined microlocal decoupling inequalities, which are related to the work of Beltran--Hickman--Sogge on Wolff-type inequalities, and an analog of Orponen's radial projection lemma which has proved quite useful in recent work on distance sets.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"144 1","pages":"1601 - 1639"},"PeriodicalIF":1.7,"publicationDate":"2019-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42530950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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