We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isometry group of a CAT$(-1)$ Hadamard manifold is strictly dominated in length spectrum by a large collection of Fuchsian ones. As an intermediate step in the proof, we obtain a result of independent interest: parametrizations of certain Teichm{"u}ller spaces by holomorphic quadratic differentials. The main consequence of the domination result is the existence of a new collection of anti-de Sitter $3$-manifolds. We also present an application to the theory of minimal immersions into the Grassmanian of timelike planes in $mathbb{R}^{2,2}$.
{"title":"Infinite energy equivariant harmonic maps, domination, and anti-de Sitter $3$-manifolds","authors":"Nathaniel Sagman","doi":"10.4310/jdg/1689262064","DOIUrl":"https://doi.org/10.4310/jdg/1689262064","url":null,"abstract":"We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. \u0000We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isometry group of a CAT$(-1)$ Hadamard manifold is strictly dominated in length spectrum by a large collection of Fuchsian ones. As an intermediate step in the proof, we obtain a result of independent interest: parametrizations of certain Teichm{\"u}ller spaces by holomorphic quadratic differentials. The main consequence of the domination result is the existence of a new collection of anti-de Sitter $3$-manifolds. We also present an application to the theory of minimal immersions into the Grassmanian of timelike planes in $mathbb{R}^{2,2}$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43155045","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}
{"title":"Codimension two holomorphic foliation","authors":"D. Cerveau, A. L. Neto","doi":"10.4310/jdg/1573786970","DOIUrl":"https://doi.org/10.4310/jdg/1573786970","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42929649","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}
Given an oriented graph on a punctured Riemann surface of arbitrary genus, we define a canonical symplectic structure over the set of flat connections on the dual graph, and show that it is invariant under natural transformations. We use this notion to identify the canonical non-degenerate extension of Goldman's symplectic form on the $SL(n)$ character variety with a the form associated to a suitable graph. Using the invariance of the form under natural moves, we utilize the parametrization of the character variety in terms of Fock--Goncharov coordinates and associate to it a canonical decorated triangulation. This allows us to show that these coordinates are log--canonical for the extended Goldman Poisson structure.
{"title":"Extended Goldman symplectic structure in Fock–Goncharov coordinates","authors":"M. Bertola, D. Korotkin","doi":"10.4310/jdg/1689262061","DOIUrl":"https://doi.org/10.4310/jdg/1689262061","url":null,"abstract":"Given an oriented graph on a punctured Riemann surface of arbitrary genus, we define a canonical symplectic structure over the set of flat connections on the dual graph, and show that it is invariant under natural transformations. We use this notion to identify the canonical non-degenerate extension of Goldman's symplectic form on the $SL(n)$ character variety with a the form associated to a suitable graph. Using the invariance of the form under natural moves, we utilize the parametrization of the character variety in terms of Fock--Goncharov coordinates and associate to it a canonical decorated triangulation. This allows us to show that these coordinates are log--canonical for the extended Goldman Poisson structure.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49014362","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}
{"title":"Convex $mathbb{RP}^2$ structures and cubic differentials under neck separation","authors":"John C. Loftin","doi":"10.4310/jdg/1571882429","DOIUrl":"https://doi.org/10.4310/jdg/1571882429","url":null,"abstract":"","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44757964","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}
In this article, we prove the Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary whose asymptotic region is modelled on a half-space. Such spaces were initially considered by Almaraz, Barbosa and de Lima in 2014. In order to prove the inequality, we develop a new approximation scheme for the weak free boundary inverse mean curvature flow, introduced by Marquardt in 2012, and establish the monotonicity of a free boundary version of the Hawking mass. Our result also implies a non-optimal Penrose inequality for asymptotically flat support surfaces in $mathbb{R}^3$ and thus sheds some light on a conjecture made by Huisken.
{"title":"The Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary","authors":"T. Koerber","doi":"10.4310/jdg/1686931603","DOIUrl":"https://doi.org/10.4310/jdg/1686931603","url":null,"abstract":"In this article, we prove the Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary whose asymptotic region is modelled on a half-space. Such spaces were initially considered by Almaraz, Barbosa and de Lima in 2014. In order to prove the inequality, we develop a new approximation scheme for the weak free boundary inverse mean curvature flow, introduced by Marquardt in 2012, and establish the monotonicity of a free boundary version of the Hawking mass. Our result also implies a non-optimal Penrose inequality for asymptotically flat support surfaces in $mathbb{R}^3$ and thus sheds some light on a conjecture made by Huisken.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47885229","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}
For a harmonic map $u:M^3to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2pi int_{thetain S^1}chi(Sigma_{theta})geq frac{1}{2}int_{thetain S^1}int_{Sigma_{theta}}(|du|^{-2}|Hess(u)|^2+R_M)$$ relating the scalar curvature $R_M$ of $M$ to the average Euler characteristic of the level sets $Sigma_{theta}=u^{-1}{theta}$. As our primary application, we extend the Kronheimer--Mrowka characterization of the Thurston norm on $H_2(M;mathbb{Z})$ in terms of $|R_M^-|_{L^2}$ and the harmonic norm to any closed $3$--manifold containing no nonseparating spheres. Additional corollaries include the Bray--Brendle--Neves rigidity theorem for the systolic inequality $(min R_M)sys_2(M)leq 8pi$, and the well--known result of Schoen and Yau that $T^3$ admits no metric of positive scalar curvature.
{"title":"Scalar curvature and harmonic maps to $S^1$","authors":"Daniel Stern","doi":"10.4310/jdg/1669998185","DOIUrl":"https://doi.org/10.4310/jdg/1669998185","url":null,"abstract":"For a harmonic map $u:M^3to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2pi int_{thetain S^1}chi(Sigma_{theta})geq frac{1}{2}int_{thetain S^1}int_{Sigma_{theta}}(|du|^{-2}|Hess(u)|^2+R_M)$$ relating the scalar curvature $R_M$ of $M$ to the average Euler characteristic of the level sets $Sigma_{theta}=u^{-1}{theta}$. As our primary application, we extend the Kronheimer--Mrowka characterization of the Thurston norm on $H_2(M;mathbb{Z})$ in terms of $|R_M^-|_{L^2}$ and the harmonic norm to any closed $3$--manifold containing no nonseparating spheres. Additional corollaries include the Bray--Brendle--Neves rigidity theorem for the systolic inequality $(min R_M)sys_2(M)leq 8pi$, and the well--known result of Schoen and Yau that $T^3$ admits no metric of positive scalar curvature.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42654624","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}
In this note we show that the Riemann moduli spaces $M_{g, n}$ equipped with the Weil--Petersson metric are quantum ergodic for $3g+n geq 4$. We also provide other examples of singular spaces with ergodic geodesic flow for which quantum ergodicity holds.
{"title":"Riemann moduli spaces are quantum ergodic","authors":"Dean Baskin, Jesse Gell-Redman, X. Han","doi":"10.4310/jdg/1683307003","DOIUrl":"https://doi.org/10.4310/jdg/1683307003","url":null,"abstract":"In this note we show that the Riemann moduli spaces $M_{g, n}$ equipped with the Weil--Petersson metric are quantum ergodic for $3g+n geq 4$. We also provide other examples of singular spaces with ergodic geodesic flow for which quantum ergodicity holds.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48304673","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}
We prove that no eigenvalue of the clamped disk can have multiplicity greater than six. Our method of proof is based on a new recursion formula, linear algebra arguments and a transcendency theorem due to Siegel and Shidlovskii.
{"title":"Bounded multiplicity for eigenvalues of a circular vibrating clamped plate","authors":"Y. Lvovsky, D. Mangoubi","doi":"10.4310/jdg/1659987895","DOIUrl":"https://doi.org/10.4310/jdg/1659987895","url":null,"abstract":"We prove that no eigenvalue of the clamped disk can have multiplicity greater than six. Our method of proof is based on a new recursion formula, linear algebra arguments and a transcendency theorem due to Siegel and Shidlovskii.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44488050","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}
It is well known that a k-dimensional smooth surface in a Euclidean space cannot be tangent to a non-involutive distribution of k-dimensional planes. In this paper we discuss the extension of this statement to weaker notions of surfaces, namely integral and normal currents. We find out that integral currents behave to this regard exactly as smooth surfaces, while the behaviour of normal currents is rather multifaceted. This issue is strictly related to a geometric property of the boundary of currents, which is also discussed in details.
{"title":"On the geometric structure of currents tangent to smooth distributions","authors":"G. Alberti, A. Massaccesi, E. Stepanov","doi":"10.4310/jdg/1668186786","DOIUrl":"https://doi.org/10.4310/jdg/1668186786","url":null,"abstract":"It is well known that a k-dimensional smooth surface in a Euclidean space cannot be tangent to a non-involutive distribution of k-dimensional planes. In this paper we discuss the extension of this statement to weaker notions of surfaces, namely integral and normal currents. We find out that integral currents behave to this regard exactly as smooth surfaces, while the behaviour of normal currents is rather multifaceted. This issue is strictly related to a geometric property of the boundary of currents, which is also discussed in details.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2019-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47939071","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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