Pub Date : 2018-07-19DOI: 10.4310/acta.2022.v228.n2.a2
Ewain Gwynne, Jason Miller, S. Sheffield
There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale varies from place to place, so that the law of the environment is in a certain sense only translation invariant modulo scaling. For our purposes, an "environment" consists of an infinite random planar map embedded in $mathbb C$, each of whose edges comes with a positive real conductance. Our main result is that under modest constraints (translation invariance modulo scaling together with the finiteness of a type of specific energy) a random walk in this kind of environment converges to Brownian motion modulo time parameterization in the quenched sense. Environments of the type considered here arise naturally in the study of random planar maps and Liouville quantum gravity. In fact, the results of this paper are used in separate works to prove that certain random planar maps (embedded in the plane via the so-called Tutte embedding) have scaling limits given by SLE-decorated Liouville quantum gravity, and also to provide a more explicit construction of Brownian motion on the Brownian map. However, the results of this paper are much more general and can be read independently of that program. One general consequence of our main result is that if a translation invariant (modulo scaling) random embedded planar map and its dual have finite energy per area, then they are close on large scales to a minimal energy embedding (the harmonic embedding). To establish Brownian motion convergence for an infinite energy embedding, it suffices to show that one can perturb it to make the energy finite.
{"title":"An invariance principle for ergodic scale-free random environments","authors":"Ewain Gwynne, Jason Miller, S. Sheffield","doi":"10.4310/acta.2022.v228.n2.a2","DOIUrl":"https://doi.org/10.4310/acta.2022.v228.n2.a2","url":null,"abstract":"There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale varies from place to place, so that the law of the environment is in a certain sense only translation invariant modulo scaling. For our purposes, an \"environment\" consists of an infinite random planar map embedded in $mathbb C$, each of whose edges comes with a positive real conductance. Our main result is that under modest constraints (translation invariance modulo scaling together with the finiteness of a type of specific energy) a random walk in this kind of environment converges to Brownian motion modulo time parameterization in the quenched sense. \u0000Environments of the type considered here arise naturally in the study of random planar maps and Liouville quantum gravity. In fact, the results of this paper are used in separate works to prove that certain random planar maps (embedded in the plane via the so-called Tutte embedding) have scaling limits given by SLE-decorated Liouville quantum gravity, and also to provide a more explicit construction of Brownian motion on the Brownian map. However, the results of this paper are much more general and can be read independently of that program. \u0000One general consequence of our main result is that if a translation invariant (modulo scaling) random embedded planar map and its dual have finite energy per area, then they are close on large scales to a minimal energy embedding (the harmonic embedding). To establish Brownian motion convergence for an infinite energy embedding, it suffices to show that one can perturb it to make the energy finite.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42123975","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 : 2018-06-01DOI: 10.4310/ACTA.2018.V220.N2.A5
B. Lamel, N. Mir
A formal holomorphic map H: (M,p)!M ′ from a germ of a real-analytic submanifold M⊂C at p∈M into a real-analytic subset M ′⊂CN ′ is an N ′-tuple of formal holomorphic power series H=(H1, ...,HN ′) satisfying H(p)∈M ′ with the property that, for any germ of a real-analytic function δ(w, w) at H(p)∈C ′ which vanishes on M ′, the formal power series δ(H(z), H(z)) vanishes on M . There is an abundance of examples showing that formal maps may diverge: After the trivial example of self-maps of a complex submanifold, possibly the simplest non-trivial example is given by the formal maps of (R, 0) into R which are just given by the formal power series in z∈C with real coefficients, that is, by elements of R[[z]]. It is a surprising fact at first that, for formal mappings between real submanifolds in complex spaces, if one assumes that the trivial examples above are excluded in a suitable sense, the situation is fundamentally different. The first result of this kind was encountered by Chern and Moser in [CM], where—as a byproduct of the convergence of their normal form—it follows that every formal holomorphic invertible map between Levinon-degenerate hypersurfaces in C necessarily converges. The convergence problem, that is, deciding whether formal maps, as described above, are in fact convergent, has been studied intensively in different contexts, both for CR manifolds and for manifolds with CR singularities, for which we refer the reader to the papers [Rot], [MMZ2], [LM1], [HY1], [HY2], [HY3], [Sto], [GS] and the references therein. Solutions to the convergence problem have important applications, for example, to the biholomorphic equivalence
{"title":"Convergence and divergence of formal CR mappings","authors":"B. Lamel, N. Mir","doi":"10.4310/ACTA.2018.V220.N2.A5","DOIUrl":"https://doi.org/10.4310/ACTA.2018.V220.N2.A5","url":null,"abstract":"A formal holomorphic map H: (M,p)!M ′ from a germ of a real-analytic submanifold M⊂C at p∈M into a real-analytic subset M ′⊂CN ′ is an N ′-tuple of formal holomorphic power series H=(H1, ...,HN ′) satisfying H(p)∈M ′ with the property that, for any germ of a real-analytic function δ(w, w) at H(p)∈C ′ which vanishes on M ′, the formal power series δ(H(z), H(z)) vanishes on M . There is an abundance of examples showing that formal maps may diverge: After the trivial example of self-maps of a complex submanifold, possibly the simplest non-trivial example is given by the formal maps of (R, 0) into R which are just given by the formal power series in z∈C with real coefficients, that is, by elements of R[[z]]. It is a surprising fact at first that, for formal mappings between real submanifolds in complex spaces, if one assumes that the trivial examples above are excluded in a suitable sense, the situation is fundamentally different. The first result of this kind was encountered by Chern and Moser in [CM], where—as a byproduct of the convergence of their normal form—it follows that every formal holomorphic invertible map between Levinon-degenerate hypersurfaces in C necessarily converges. The convergence problem, that is, deciding whether formal maps, as described above, are in fact convergent, has been studied intensively in different contexts, both for CR manifolds and for manifolds with CR singularities, for which we refer the reader to the papers [Rot], [MMZ2], [LM1], [HY1], [HY2], [HY3], [Sto], [GS] and the references therein. Solutions to the convergence problem have important applications, for example, to the biholomorphic equivalence","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"220 1","pages":"367-406"},"PeriodicalIF":3.7,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42862607","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 : 2018-06-01DOI: 10.4310/ACTA.2018.V220.N2.A2
Serge Cantat, Junyi Xie
We study groups of automorphisms and birational transformations of quasi-projective varieties. Two methods are combined; the first one is based on p-adic analysis, the second makes use of isoperimetric inequalities and LangWeil estimates. For instance, we show that if SL n(Z) acts faithfully on a complex quasi-projective variety X by birational transformations, then dim(X) ≥ n−1 and X is rational if dim(X) = n−1.
{"title":"Algebraic actions of discrete groups: the $p$-adic method","authors":"Serge Cantat, Junyi Xie","doi":"10.4310/ACTA.2018.V220.N2.A2","DOIUrl":"https://doi.org/10.4310/ACTA.2018.V220.N2.A2","url":null,"abstract":"We study groups of automorphisms and birational transformations of quasi-projective varieties. Two methods are combined; the first one is based on p-adic analysis, the second makes use of isoperimetric inequalities and LangWeil estimates. For instance, we show that if SL n(Z) acts faithfully on a complex quasi-projective variety X by birational transformations, then dim(X) ≥ n−1 and X is rational if dim(X) = n−1.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"220 1","pages":"239-295"},"PeriodicalIF":3.7,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48028557","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 : 2018-05-20DOI: 10.4310/ACTA.2021.v226.n1.a1
Paul D. Nelson, Akshay Venkatesh
We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. Our main global application is an asymptotic formula for averages of Gan--Gross--Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding $L$-functions. Ratner's results on measure classification provide an important input to the proof. Our local results include asymptotic expansions for certain special functions arising from representations of higher rank Lie groups, such as the relative characters defined by matrix coefficient integrals as in the Ichino--Ikeda conjecture.
{"title":"The orbit method and analysis of automorphic forms","authors":"Paul D. Nelson, Akshay Venkatesh","doi":"10.4310/ACTA.2021.v226.n1.a1","DOIUrl":"https://doi.org/10.4310/ACTA.2021.v226.n1.a1","url":null,"abstract":"We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. \u0000Our main global application is an asymptotic formula for averages of Gan--Gross--Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding $L$-functions. Ratner's results on measure classification provide an important input to the proof. \u0000Our local results include asymptotic expansions for certain special functions arising from representations of higher rank Lie groups, such as the relative characters defined by matrix coefficient integrals as in the Ichino--Ikeda conjecture.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48408084","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 : 2018-01-23DOI: 10.4310/ACTA.2019.V222.N2.A2
D. Bucur, A. Henrot
In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{'o}lya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.
{"title":"Maximization of the second non-trivial Neumann eigenvalue","authors":"D. Bucur, A. Henrot","doi":"10.4310/ACTA.2019.V222.N2.A2","DOIUrl":"https://doi.org/10.4310/ACTA.2019.V222.N2.A2","url":null,"abstract":"In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{'o}lya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41510317","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 : 2018-01-04DOI: 10.4310/ACTA.2019.V222.N2.A1
Chiara Boccato, C. Brennecke, S. Cenatiempo, B. Schlein
We consider Bose gases consisting of $N$ particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of the order $N^{-1}$(Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as $N to infty$. Our results confirm Bogoliubov's predictions.
{"title":"Bogoliubov theory in the Gross–Pitaevskii limit","authors":"Chiara Boccato, C. Brennecke, S. Cenatiempo, B. Schlein","doi":"10.4310/ACTA.2019.V222.N2.A1","DOIUrl":"https://doi.org/10.4310/ACTA.2019.V222.N2.A1","url":null,"abstract":"We consider Bose gases consisting of $N$ particles trapped in a box with volume one and interacting through a repulsive potential with scattering length of the order $N^{-1}$(Gross-Pitaevskii regime). We determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing as $N to infty$. Our results confirm Bogoliubov's predictions.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45255097","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 : 2018-01-01DOI: 10.4310/acta.2018.v221.n2.a1
Thomas Nikolaus, Peter Scholze
Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by Bokstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas--Goodwillie--McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing $K$-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum. The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the $infty$-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum $X$ with $S^1$-action (in the most naive sense) together with $S^1$-equivariant maps $varphi_p: Xto X^{tC_p}$ for all primes $p$. Here $X^{tC_p}=mathrm{cofib}(mathrm{Nm}: X_{hC_p}to X^{hC_p})$ is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology. In order to construct the maps $varphi_p: Xto X^{tC_p}$ in the example of topological Hochschild homology we introduce and study Tate diagonals for spectra and Frobenius homomorphisms of commutative ring spectra. In particular we prove a version of the Segal conjecture for the Tate diagonals and relate these Frobenius homomorphisms to power operations.
{"title":"On topological cyclic homology","authors":"Thomas Nikolaus, Peter Scholze","doi":"10.4310/acta.2018.v221.n2.a1","DOIUrl":"https://doi.org/10.4310/acta.2018.v221.n2.a1","url":null,"abstract":"Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by Bokstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas--Goodwillie--McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing $K$-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum. The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the $infty$-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum $X$ with $S^1$-action (in the most naive sense) together with $S^1$-equivariant maps $varphi_p: Xto X^{tC_p}$ for all primes $p$. Here $X^{tC_p}=mathrm{cofib}(mathrm{Nm}: X_{hC_p}to X^{hC_p})$ is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology. In order to construct the maps $varphi_p: Xto X^{tC_p}$ in the example of topological Hochschild homology we introduce and study Tate diagonals for spectra and Frobenius homomorphisms of commutative ring spectra. In particular we prove a version of the Segal conjecture for the Tate diagonals and relate these Frobenius homomorphisms to power operations.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"5 22","pages":""},"PeriodicalIF":3.7,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494786","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 : 2017-12-22DOI: 10.4310/ACTA.2019.V222.N2.A3
Y. Shitov
The multiplicative complexity of systems of bilinear forms (and, in particular, the famous question of fast matrix multiplication) is an important area of research in modern theory of computation. One of the foundational papers on the topic is Strassen’s work [20], which contains an O(n 7/ ln ) algorithm for the multiplication of two n×n matrices. In his subsequent paper [21] published in 1973, Strassen asked whether the multiplicative complexity of the union of two bilinear systems depending on different variables is equal to the sum of the multiplicative complexities of both systems. A stronger version of this problem was proposed in the 1981 paper [10] by Feig and Winograd, who asked whether any optimal algorithm that computes such a pair of bilinear systems must compute each system separately. These questions became known as the direct sum conjecture and strong direct sum conjecture, respectively, and they were attracting a notable amount of attention during the four decades. As Feig and Winograd wrote, ‘either a proof of, or a counterexample to, the direct sum conjecture will be a major step forward in our understanding of complexity of systems of bilinear forms.’ The modern formulation of this conjecture is based on a natural representation of a bilinear system as a three-dimensional tensor, that is, an array of elements T (i|j|k) taken from a field F , where the triples (i, j, k) run over the Cartesian product of finite indexing sets I, J,K. A tensor T is called decomposable if T = a⊗b⊗c (which should be read as T (i|j|k) = aibjck), for some vectors a ∈ FI , b ∈ FJ , c ∈ FK . The rank of a tensor T , or the multiplicative complexity of the corresponding bilinear system, is the smallest r for which T can be written as a sum of r decomposable tensors with entries in F . We denote this quantity by rankF T , and we note that the rank of a tensor may change if one allows to take the entries of decomposable tensors as above from an extension of F , see [3]. Taking the union of two bilinear systems depending on disjoint sets of variables corresponds to the direct sum operation on tensors. More precisely, if T and T ′ are tensors with disjoint indexing sets I, I , J, J ,K,K , then we can define the direct sum T⊕T ′ as a tensor with indexing sets I ∪ I , J ∪ J , K ∪ K ′ such that the (I|J |K) block equals T and (I ′|J ′|K ) block equals T , and all entries outside of these blocks are zero. In other words, direct sums of tensors are a multidimensional analogue of block-diagonal matrices; a basic result of linear algebra says that the ranks of such matrices are equal to the sums of the ranks of their diagonal blocks. Strassen’s direct sum conjecture is a three-dimensional analogue of this statement.
{"title":"Counterexamples to Strassen’s direct sum conjecture","authors":"Y. Shitov","doi":"10.4310/ACTA.2019.V222.N2.A3","DOIUrl":"https://doi.org/10.4310/ACTA.2019.V222.N2.A3","url":null,"abstract":"The multiplicative complexity of systems of bilinear forms (and, in particular, the famous question of fast matrix multiplication) is an important area of research in modern theory of computation. One of the foundational papers on the topic is Strassen’s work [20], which contains an O(n 7/ ln ) algorithm for the multiplication of two n×n matrices. In his subsequent paper [21] published in 1973, Strassen asked whether the multiplicative complexity of the union of two bilinear systems depending on different variables is equal to the sum of the multiplicative complexities of both systems. A stronger version of this problem was proposed in the 1981 paper [10] by Feig and Winograd, who asked whether any optimal algorithm that computes such a pair of bilinear systems must compute each system separately. These questions became known as the direct sum conjecture and strong direct sum conjecture, respectively, and they were attracting a notable amount of attention during the four decades. As Feig and Winograd wrote, ‘either a proof of, or a counterexample to, the direct sum conjecture will be a major step forward in our understanding of complexity of systems of bilinear forms.’ The modern formulation of this conjecture is based on a natural representation of a bilinear system as a three-dimensional tensor, that is, an array of elements T (i|j|k) taken from a field F , where the triples (i, j, k) run over the Cartesian product of finite indexing sets I, J,K. A tensor T is called decomposable if T = a⊗b⊗c (which should be read as T (i|j|k) = aibjck), for some vectors a ∈ FI , b ∈ FJ , c ∈ FK . The rank of a tensor T , or the multiplicative complexity of the corresponding bilinear system, is the smallest r for which T can be written as a sum of r decomposable tensors with entries in F . We denote this quantity by rankF T , and we note that the rank of a tensor may change if one allows to take the entries of decomposable tensors as above from an extension of F , see [3]. Taking the union of two bilinear systems depending on disjoint sets of variables corresponds to the direct sum operation on tensors. More precisely, if T and T ′ are tensors with disjoint indexing sets I, I , J, J ,K,K , then we can define the direct sum T⊕T ′ as a tensor with indexing sets I ∪ I , J ∪ J , K ∪ K ′ such that the (I|J |K) block equals T and (I ′|J ′|K ) block equals T , and all entries outside of these blocks are zero. In other words, direct sums of tensors are a multidimensional analogue of block-diagonal matrices; a basic result of linear algebra says that the ranks of such matrices are equal to the sums of the ranks of their diagonal blocks. Strassen’s direct sum conjecture is a three-dimensional analogue of this statement.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2017-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46675899","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 : 2017-12-19DOI: 10.4310/ACTA.2020.v224.n1.a1
David Bate
We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $fcolon X to mathbb R^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of all $f$ with $mathcal H^n(f(S))=0$ is residual. Conversely, if $Esubset X$ is $n$-rectifiable with $mathcal H^n(E)>0$, the set of all $f$ with $mathcal H^n(f(E))>0$ is residual. These results provide a replacement for the Besicovitch-Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.
{"title":"Purely unrectifiable metric spaces and perturbations of Lipschitz functions","authors":"David Bate","doi":"10.4310/ACTA.2020.v224.n1.a1","DOIUrl":"https://doi.org/10.4310/ACTA.2020.v224.n1.a1","url":null,"abstract":"We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $fcolon X to mathbb R^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of all $f$ with $mathcal H^n(f(S))=0$ is residual. Conversely, if $Esubset X$ is $n$-rectifiable with $mathcal H^n(E)>0$, the set of all $f$ with $mathcal H^n(f(E))>0$ is residual. \u0000These results provide a replacement for the Besicovitch-Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":" ","pages":""},"PeriodicalIF":3.7,"publicationDate":"2017-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46725174","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 : 2017-11-24DOI: 10.4467/20843828am.17.001.7077
P. Åhag, R. Czyż
{"title":"On the complex Monge-Ampère operator in unbounded domains","authors":"P. Åhag, R. Czyż","doi":"10.4467/20843828am.17.001.7077","DOIUrl":"https://doi.org/10.4467/20843828am.17.001.7077","url":null,"abstract":"","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":"2017 1","pages":"7-13"},"PeriodicalIF":3.7,"publicationDate":"2017-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48900290","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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