Analysis & PDE最新文献

{"title":"Haagerup’s phase transition at polydisc slicing","authors":"Giorgos Chasapis, Salil Singh, Tomasz Tkocz","doi":"10.2140/apde.2024.17.2509","DOIUrl":"https://doi.org/10.2140/apde.2024.17.2509","url":null,"abstract":"<p>We establish a sharp comparison inequality between the negative moments and the second moment of the magnitude of sums of independent random vectors uniform on three-dimensional Euclidean spheres. This provides a probabilistic extension of the Oleszkiewicz–Pełczyński polydisc slicing result. The Haagerup-type phase transition occurs exactly when the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi></math>-norm recovers volume, in contrast to the real case. We also obtain partial results in higher dimensions. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217537","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":"Uniform Skoda integrability and Calabi–Yau degeneration","authors":"Yang Li","doi":"10.2140/apde.2024.17.2247","DOIUrl":"https://doi.org/10.2140/apde.2024.17.2247","url":null,"abstract":"<p>We study polarised algebraic degenerations of Calabi–Yau manifolds. We prove a uniform Skoda-type estimate and a uniform <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math>-estimate for the Calabi–Yau Kähler potentials. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217528","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":"Nonlinear periodic waves on the Einstein cylinder","authors":"Athanasios Chatzikaleas, Jacques Smulevici","doi":"10.2140/apde.2024.17.2311","DOIUrl":"https://doi.org/10.2140/apde.2024.17.2311","url":null,"abstract":"<p>Motivated by the study of small amplitude nonlinear waves in the anti-de Sitter spacetime and in particular the conjectured existence of periodic in time solutions to the Einstein equations, we construct families of arbitrary small time-periodic solutions to the conformal cubic wave equation and the spherically symmetric Yang–Mills equations on the Einstein cylinder <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℝ</mi>\u0000<mo>×</mo> <msup><mrow><mi mathvariant=\"double-struck\">𝕊</mi></mrow><mrow><mn>3</mn></mrow></msup></math>. For the conformal cubic wave equation, we consider both spherically symmetric solutions and complex-valued aspherical solutions with an ansatz relying on the Hopf fibration of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math>-sphere. In all three cases, the equations reduce to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math>+<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>1</mn></math> semilinear wave equations. </p><p> Our proof relies on a theorem of Bambusi–Paleari for which the main assumption is the existence of a seed solution, given by a nondegenerate zero of a nonlinear operator associated with the resonant system. For the problems that we consider, such seed solutions are simply given by the mode solutions of the linearized equations. Provided that the Fourier coefficients of the systems can be computed, the nondegeneracy conditions then amount to solving infinite dimensional linear systems. Since the eigenfunctions for all three cases studied are given by Jacobi polynomials, we derive the different Fourier and resonant systems using linearization and connection formulas as well as integral transformation of Jacobi polynomials. </p><p> In the Yang–Mills case, the original version of the theorem of Bambusi–Paleari is not applicable because the nonlinearity of smallest degree is nonresonant. The resonant terms are then provided by the next order nonlinear terms with an extra correction due to backreaction terms of the smallest degree of nonlinearity, and we prove an analogous theorem in this setting. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217543","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":"Unique continuation for the heat operator with potentials in weak spaces","authors":"Eunhee Jeong, Sanghyuk Lee, Jaehyeon Ryu","doi":"10.2140/apde.2024.17.2257","DOIUrl":"https://doi.org/10.2140/apde.2024.17.2257","url":null,"abstract":"<p>We prove the strong unique continuation property for the differential inequality </p>\u0000<div><math display=\"block\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mo>|</mo><mo stretchy=\"false\">(</mo><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub>\u0000<mo>+</mo> <mi mathvariant=\"normal\">Δ</mi><mo stretchy=\"false\">)</mo><mi>u</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo>|</mo><mo>≤</mo>\u0000<mi>V</mi>\u0000<mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo>|</mo><mi>u</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo>|</mo><mo>,</mo>\u0000</math>\u0000</div>\u0000<p> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi> </math> contained in weak spaces. In particular, we establish the strong unique continuation property for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi>\u0000<mo>∈</mo> <msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>∞</mi></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>x</mi></mrow><mrow><mo stretchy=\"false\">[</mo><mi>t</mi><mo stretchy=\"false\">]</mo><mi>d</mi><mo>∕</mo><mn>2</mn><mo>,</mo><mi>∞</mi></mrow></msubsup></math>, which has been left open since the works of Escauriaza (2000) and Escauriaza and Vega (2001). Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217535","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":"Beurling–Carleson sets, inner functions and a semilinear equation","authors":"Oleg Ivrii, Artur Nicolau","doi":"10.2140/apde.2024.17.2585","DOIUrl":"https://doi.org/10.2140/apde.2024.17.2585","url":null,"abstract":"&lt;p&gt;Beurling–Carleson sets have appeared in a number of areas of complex analysis such as boundary zero sets of analytic functions, inner functions with derivative in the Nevanlinna class, cyclicity in weighted Bergman spaces, Fuchsian groups of Widom-type and the corona problem in quotient Banach algebras. After surveying these developments, we give a general definition of Beurling–Carleson sets and discuss some of their basic properties. We show that the Roberts decomposition characterizes measures that do not charge Beurling–Carleson sets. &lt;/p&gt;&lt;p&gt; For a positive singular measure &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/math&gt; on the unit circle, let &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; denote the singular inner function with singular measure &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/math&gt;. In the second part of the paper, we use a corona-type decomposition to relate a number of properties of singular measures on the unit circle, such as membership of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;′&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt; in the Nevanlinna class &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi mathvariant=\"bold-script\"&gt;𝒩&lt;/mi&gt;&lt;/math&gt;, area conditions on level sets of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; and wepability. It was known that each of these properties holds for measures concentrated on Beurling–Carleson sets. We show that each of these properties implies that &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/math&gt; lives on a countable union of Beurling–Carleson sets. We also describe partial relations involving the membership of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;′&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt; in the Hardy space &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;, membership of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt; in the Besov space &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; and &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;-Beurling–Carleson sets and give a number of examples which show that our results are optimal. &lt;/p&gt;&lt;p&gt; Finally, we show that measures that live on countable unions of &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/math&gt;-Beurl","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217540","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":"A fast point charge interacting with the screened Vlasov–Poisson system","authors":"Richard M. Höfer, Raphael Winter","doi":"10.2140/apde.2024.17.2451","DOIUrl":"https://doi.org/10.2140/apde.2024.17.2451","url":null,"abstract":"<p>We consider the long-time behavior of a fast, charged particle interacting with an initially spatially homogeneous background plasma. The background is modeled by the screened Vlasov–Poisson equations, whereas the interaction potential of the point charge is assumed to be smooth. We rigorously prove the validity of the <span>stopping power theory </span>in physics, which predicts a decrease of the velocity <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>V</mi>\u0000<mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo></math> of the point charge given by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mover accent=\"true\"><mrow><mi>V</mi> </mrow><mo accent=\"true\">˙</mo></mover>\u0000<mo>∼</mo><mo>−</mo><mo>|</mo><mi>V</mi> <msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup><mi>V</mi></math>, a formula that goes back to Bohr (1915). Our result holds for all initial velocities larger than a threshold value that is larger than the velocity of all background particles and remains valid until the particle slows down to the threshold velocity or the time is exponentially long compared to the velocity of the point charge. </p><p> The long-time behavior of this coupled system is related to the question of Landau damping, which has remained open in this setting so far. Contrary to other results in nonlinear Landau damping, the long-time behavior of the system is driven by the nontrivial electric field of the plasma, and the damping only occurs in regions that the point charge has already passed. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217541","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":"Host–Kra factors for ⊕ p∈Pℤ∕pℤ actions and finite-dimensional nilpotent systems","authors":"Or Shalom","doi":"10.2140/apde.2024.17.2379","DOIUrl":"https://doi.org/10.2140/apde.2024.17.2379","url":null,"abstract":"<p>Let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">𝒫</mi></math> be a countable multiset of primes and let <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi>\u0000<mo>=</mo><msub><mrow><mi> ⊕</mi><mo> ⁡<!--FUNCTION APPLICATION--></mo>\u0000<!--nolimits--></mrow><mrow><mi>p</mi><mo>∈</mo><mi>P</mi></mrow></msub><mi>ℤ</mi><mo>∕</mo><mi>p</mi><mi>ℤ</mi></math>. We study the universal characteristic factors associated with the Gowers–Host–Kra seminorms for the group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math>. We show that the universal characteristic factor of order <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\u0000<mo>&lt;</mo>\u0000<mi>k</mi>\u0000<mo>+</mo> <mn>1</mn></math> is a factor of an inverse limit of <span>finite-dimensional</span>\u0000<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math><span>-step</span>\u0000<span>nilpotent homogeneous spaces</span>. The latter is a counterpart of a <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>-step nilsystem where the homogeneous group is not necessarily a Lie group. As an application of our structure theorem we derive an alternative proof for the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math>-convergence of multiple ergodic averages associated with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>-term arithmetic progressions in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>G</mi></math> and derive a formula for the limit in the special case where the underlying space is a nilpotent homogeneous system. Our results provide a counterpart of the structure theorem of Host and Kra (2005) and Ziegler (2007) concerning <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℤ</mi></math>-actions and generalize the results of Bergelson, Tao and Ziegler (2011, 2015) concerning <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>ω</mi></mrow></msubsup></math>-actions. This is also the first instance of studying the Host–Kra factors of nonfinitely generated groups of unbounded torsion. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217539","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":"Trigonometric chaos and Xp inequalities, I : Balanced Fourier truncations over discrete groups","authors":"Antonio Ismael Cano-Mármol, José M. Conde-Alonso, Javier Parcet","doi":"10.2140/apde.2024.17.2561","DOIUrl":"https://doi.org/10.2140/apde.2024.17.2561","url":null,"abstract":"<p>We investigate <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math>-estimates for balanced averages of Fourier truncations in group algebras, in terms of “differential operators” acting on them. Our results extend a fundamental inequality of Naor for the hypercube (with profound consequences in metric geometry) to discrete groups. Different inequalities are established in terms of “directional derivatives” which are constructed via affine representations determined by the Fourier truncations. Our proofs rely on the Banach <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> X</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>p</mi></mrow></msub></math> nature of noncommutative <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math>-spaces and dimension-free estimates for noncommutative Riesz transforms. In the particular case of free groups we use an alternative approach based on free Hilbert transforms. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217542","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":"A substitute for Kazhdan’s property (T) for universal nonlattices","authors":"Narutaka Ozawa","doi":"10.2140/apde.2024.17.2541","DOIUrl":"https://doi.org/10.2140/apde.2024.17.2541","url":null,"abstract":"<p>The well-known theorem of Shalom–Vaserstein and Ershov–Jaikin-Zapirain states that the group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> EL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi mathvariant=\"bold-script\">ℛ</mi><mo stretchy=\"false\">)</mo></math>, generated by elementary matrices over a finitely generated commutative ring <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">ℛ</mi></math>, has Kazhdan’s property (T) as soon as <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\u0000<mo>≥</mo> <mn>3</mn></math>. This is no longer true if the ring <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"bold-script\">ℛ</mi></math> is replaced by a commutative rng (a ring but without the identity) due to nilpotent quotients <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> EL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi mathvariant=\"bold-script\">ℛ</mi><mo>∕</mo><msup><mrow><mi mathvariant=\"bold-script\">ℛ</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy=\"false\">)</mo></math>. We prove that even in such a case the group <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> EL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi mathvariant=\"bold-script\">ℛ</mi><mo stretchy=\"false\">)</mo></math> satisfies a certain property that can substitute property (T), provided that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> is large enough. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217538","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":"Nonnegative Ricci curvature and minimal graphs with linear growth","authors":"Giulio Colombo, Eddygledson S. Gama, Luciano Mari, Marco Rigoli","doi":"10.2140/apde.2024.17.2275","DOIUrl":"https://doi.org/10.2140/apde.2024.17.2275","url":null,"abstract":"<p>We study minimal graphs with linear growth on complete manifolds <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>M</mi></mrow><mrow><mi>m</mi></mrow></msup></math> with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> Ric</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits-->\u0000<mo>≥</mo> <mn>0</mn></math>. Under the further assumption that the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mi>m</mi><mo>−</mo><mn>2</mn><mo stretchy=\"false\">)</mo></math>-th Ricci curvature in radial direction is bounded below by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>C</mi><mi>r</mi><msup><mrow><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></math>, we prove that any such graph, if nonconstant, forces tangent cones at infinity of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> to split off a line. Note that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> is not required to have Euclidean volume growth. We also show that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>M</mi></math> may not split off any line. Our result parallels that obtained by Cheeger, Colding and Minicozzi for harmonic functions. The core of the paper is a new refinement of Korevaar’s gradient estimate for minimal graphs, together with heat equation techniques. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217536","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}