Pub Date : 2020-01-13DOI: 10.4310/cjm.2023.v11.n2.a1
N. Kapouleas, Peter J. McGrath
In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $Sigma$ is a closed minimal surface embedded in a Riemannian three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a suitable family of LD solutions on $Sigma$, a minimal surface $breve{M}$ resembling two copies of $Sigma$ joined by many small catenoidal bridges can be constructed by PDE gluing methods. (An LD solution $varphi$ on $Sigma$ is a singular solution of the Jacobi equation with logarithmic singularities which in the construction are replaced by catenoidal bridges.) We also determine the first nontrivial term in the expansion for the area $|breve{M}|$ of $breve{M}$ in terms of the sizes of its catenoidal bridges and confirm that it is negative; $|breve{M}|<2 | Sigma|$ follows. We demonstrate the applicability of the theorem by first constructing new doublings of the Clifford torus. We then construct in Part II families of LD solutions for general $(O(2)times mathbb{Z}_2)$-symmetric backgrounds $(Sigma, N,g)$. Combining with the theorem in Part I this implies the construction of new minimal doublings for such backgrounds. (Constructions for general backgrounds remain open.) This generalizes our earlier work for $Sigma=mathbb{S}^2 subset N=mathbb{S}^3$ providing new constructions even in that case. In Part III, applying the results of Parts I and II -- appropriately modified for the catenoid and the critical catenoid -- we construct new self-shrinkers of the mean curvature flow via doubling the spherical self-shrinker or the Angenent torus, new complete embedded minimal surfaces of finite total curvature in the Euclidean three-space via doubling the catenoid, and new free boundary minimal surfaces in the unit ball via doubling the critical catenoid.
{"title":"Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers","authors":"N. Kapouleas, Peter J. McGrath","doi":"10.4310/cjm.2023.v11.n2.a1","DOIUrl":"https://doi.org/10.4310/cjm.2023.v11.n2.a1","url":null,"abstract":"In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $Sigma$ is a closed minimal surface embedded in a Riemannian three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a suitable family of LD solutions on $Sigma$, a minimal surface $breve{M}$ resembling two copies of $Sigma$ joined by many small catenoidal bridges can be constructed by PDE gluing methods. (An LD solution $varphi$ on $Sigma$ is a singular solution of the Jacobi equation with logarithmic singularities which in the construction are replaced by catenoidal bridges.) We also determine the first nontrivial term in the expansion for the area $|breve{M}|$ of $breve{M}$ in terms of the sizes of its catenoidal bridges and confirm that it is negative; $|breve{M}|<2 | Sigma|$ follows. We demonstrate the applicability of the theorem by first constructing new doublings of the Clifford torus. We then construct in Part II families of LD solutions for general $(O(2)times mathbb{Z}_2)$-symmetric backgrounds $(Sigma, N,g)$. Combining with the theorem in Part I this implies the construction of new minimal doublings for such backgrounds. (Constructions for general backgrounds remain open.) This generalizes our earlier work for $Sigma=mathbb{S}^2 subset N=mathbb{S}^3$ providing new constructions even in that case. In Part III, applying the results of Parts I and II -- appropriately modified for the catenoid and the critical catenoid -- we construct new self-shrinkers of the mean curvature flow via doubling the spherical self-shrinker or the Angenent torus, new complete embedded minimal surfaces of finite total curvature in the Euclidean three-space via doubling the catenoid, and new free boundary minimal surfaces in the unit ball via doubling the critical catenoid.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41549152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.4310/cjm.2020.v8.n2.a1
B. Andrews, J. Clutterbuck, Daniel Hauer
On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this paper we show that this is false, by analysing the perturbation problem from the Neumann case. In particular we prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation of the ground state with respect to the Robin parameter is not a concave function. We conclude from this that the Robin ground state is not log-concave (and indeed even has some superlevel sets which are non-convex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on arbitrary convex domains which approximate these in Hausdorff distance.
{"title":"Non-concavity of the Robin ground state","authors":"B. Andrews, J. Clutterbuck, Daniel Hauer","doi":"10.4310/cjm.2020.v8.n2.a1","DOIUrl":"https://doi.org/10.4310/cjm.2020.v8.n2.a1","url":null,"abstract":"On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this paper we show that this is false, by analysing the perturbation problem from the Neumann case. In particular we prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation of the ground state with respect to the Robin parameter is not a concave function. We conclude from this that the Robin ground state is not log-concave (and indeed even has some superlevel sets which are non-convex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on arbitrary convex domains which approximate these in Hausdorff distance.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70404457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-10DOI: 10.4310/cjm.2022.v10.n1.a3
B. Collins, A. Guionnet, Félix Parraud
Let $X^N = (X_1^N,dots, X^N_d)$ be a d-tuple of $Ntimes N$ independent GUE random matrices and $Z^{NM}$ be any family of deterministic matrices in $mathbb{M}_N(mathbb{C})otimes mathbb{M}_M(mathbb{C})$. Let $P$ be a self-adjoint non-commutative polynomial. A seminal work of Voiculescu shows that the empirical measure of the eigenvalues of $P(X^N)$ converges towards a deterministic measure defined thanks to free probability theory. Let now $f$ be a smooth function, the main technical result of this paper is a precise bound of the difference between the expectation of $$frac{1}{MN}text{Tr}left( f(P(X^Notimes I_M,Z^{NM})) right)$$ and its limit when $N$ goes to infinity. If $f$ is six times differentiable, we show that it is bounded by $M^2leftVert frightVert_{mathcal{C}^6}N^{-2}$. As a corollary we obtain a new proof of a result of Haagerup and Thorbjo rnsen, later developed by Male, which gives sufficient conditions for the operator norm of a polynomial evaluated in $(X^N,Z^{NM},{Z^{NM}}^*)$ to converge almost surely towards its free limit. Restricting ourselves to polynomials in independent GUE matrices, we give concentration estimates on the largest eingenvalue of these polynomials around their free limit. A direct consequence of these inequalities is that there exists some $beta>0$ such that for any $varepsilon_1<3+beta)^{-1}$ and $varepsilon_2<1/4$, almost surely for $N$ large enough, $$-frac{1}{N^{varepsilon_1}} leq | P(X^N)| - leftVert P(x)rightVert leq frac{1}{N^{varepsilon_2}}.$$ Finally if $X^N$ and $Y^{M_N}$ are independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any polynomial in $(X^Notimes I_{M_N},I_Notimes Y^{M_N})$ converges almost surely towards its free limit. This result is an improvement of a Theorem of Pisier, who was himself using estimates from Haagerup and Thorbjo rnsen, where $M_N$ had size $o(N^{1/4})$.
设$X^N=（X_1^N，dots，X^N_d）$是$NtimesN$独立GUE随机矩阵的d元组，$Z^｛NM｝$是$mathbb中的任何确定性矩阵族{M}_N（mathbb｛C｝）otimesmathbb{M}_M（mathbb｛C｝）$。设$P$是一个自伴非交换多项式。Voiculescu的一项开创性工作表明，$P（X^N）$的特征值的经验测度收敛于自由概率论定义的确定性测度。假设$f$是一个光滑函数，本文的主要技术结果是$$frac｛1｝｛MN｝text｛Tr｝left（f（P（X^Notimes I_M，Z^｛NM｝））$$的期望值与其在$N$无穷大时的极限值之差的精确界。如果$f$是六次可微的，我们证明它有界于$M^2 left Vert fright Vert_｛mathcal｛C｝^6｝N^｛-2｝$。作为推论，我们得到了Haagerup和Thorbjo-rnsen结果的一个新证明，该结果后来由Male发展，它给出了在$（X^N，Z^{NM}，{Z^{NM}}^*）$中评估的多项式的算子范数几乎肯定地收敛于其自由极限的充分条件。将我们自己限制在独立GUE矩阵中的多项式上，我们给出了这些多项式在其自由极限附近的最大生成值的集中估计。这些不等式的一个直接结果是，存在一些$beta>0$，使得对于任何$varepsilon_1<3+beta）^｛-1｝$和$varepilon_2<1/4$，几乎可以肯定的是，对于足够大的$N$，$$-frac｛1｝｛N^｛varepsillon_1｝｝leq|P（X^N）|-leftVert P（X）rightVertleqfrac{1｝最后，如果$X^N$和$Y^{M_N}$是独立的，并且$M_N=o（N^{1/3}）$，那么几乎可以肯定的是，$（X^Notimes I_{M_N}，I_Notime Y^{M.N}）$中任何多项式的范数几乎可以肯定地收敛于其自由极限。这一结果是对Pisier定理的改进，Pisier自己使用了Haagerup和Thorbjo-rnsen的估计，其中$M_N$的大小为$o（N^{1/4}）$。
{"title":"On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices","authors":"B. Collins, A. Guionnet, Félix Parraud","doi":"10.4310/cjm.2022.v10.n1.a3","DOIUrl":"https://doi.org/10.4310/cjm.2022.v10.n1.a3","url":null,"abstract":"Let $X^N = (X_1^N,dots, X^N_d)$ be a d-tuple of $Ntimes N$ independent GUE random matrices and $Z^{NM}$ be any family of deterministic matrices in $mathbb{M}_N(mathbb{C})otimes mathbb{M}_M(mathbb{C})$. Let $P$ be a self-adjoint non-commutative polynomial. A seminal work of Voiculescu shows that the empirical measure of the eigenvalues of $P(X^N)$ converges towards a deterministic measure defined thanks to free probability theory. Let now $f$ be a smooth function, the main technical result of this paper is a precise bound of the difference between the expectation of $$frac{1}{MN}text{Tr}left( f(P(X^Notimes I_M,Z^{NM})) right)$$ and its limit when $N$ goes to infinity. If $f$ is six times differentiable, we show that it is bounded by $M^2leftVert frightVert_{mathcal{C}^6}N^{-2}$. As a corollary we obtain a new proof of a result of Haagerup and Thorbjo rnsen, later developed by Male, which gives sufficient conditions for the operator norm of a polynomial evaluated in $(X^N,Z^{NM},{Z^{NM}}^*)$ to converge almost surely towards its free limit. Restricting ourselves to polynomials in independent GUE matrices, we give concentration estimates on the largest eingenvalue of these polynomials around their free limit. A direct consequence of these inequalities is that there exists some $beta>0$ such that for any $varepsilon_1<3+beta)^{-1}$ and $varepsilon_2<1/4$, almost surely for $N$ large enough, $$-frac{1}{N^{varepsilon_1}} leq | P(X^N)| - leftVert P(x)rightVert leq frac{1}{N^{varepsilon_2}}.$$ Finally if $X^N$ and $Y^{M_N}$ are independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any polynomial in $(X^Notimes I_{M_N},I_Notimes Y^{M_N})$ converges almost surely towards its free limit. This result is an improvement of a Theorem of Pisier, who was himself using estimates from Haagerup and Thorbjo rnsen, where $M_N$ had size $o(N^{1/4})$.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42366093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-11-08DOI: 10.4310/cjm.2023.v11.n2.a2
C. Chan, A. Ivanov
For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from the classical Deligne--Lusztig theory by using the loop space functor. We study this construction in special the case that $G$ is an inner form of ${rm GL}_n$ and the loop Deligne--Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its $ell$-adic cohomology realizes many irreducible supercuspidal representations of $G$, notably almost all among those whose L-parameter factors through an unramified elliptic maximal torus of $G$. This gives a purely local, purely geometric and -- in a sense -- quite explicit way to realize special cases of the local Langlands and Jacquet--Langlands correspondences.
{"title":"On loop Deligne–Lusztig varieties of Coxeter-type for inner forms of $mathrm{GL}_n$","authors":"C. Chan, A. Ivanov","doi":"10.4310/cjm.2023.v11.n2.a2","DOIUrl":"https://doi.org/10.4310/cjm.2023.v11.n2.a2","url":null,"abstract":"For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from the classical Deligne--Lusztig theory by using the loop space functor. We study this construction in special the case that $G$ is an inner form of ${rm GL}_n$ and the loop Deligne--Lusztig variety is of Coxeter type. After simplifying the proof of its representability, our main result is that its $ell$-adic cohomology realizes many irreducible supercuspidal representations of $G$, notably almost all among those whose L-parameter factors through an unramified elliptic maximal torus of $G$. This gives a purely local, purely geometric and -- in a sense -- quite explicit way to realize special cases of the local Langlands and Jacquet--Langlands correspondences.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48571934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-30DOI: 10.4310/cjm.2021.v9.n4.a4
T. Elgindi, T. Ghoul, N. Masmoudi
We study the stability of recently constructed self-similar blow-up solutions to the incompressible Euler equation. A consequence of our work is the existence of finite-energy $C^{1,alpha}$ solutions that become singular in finite time in a locally self-similar manner. As a corollary, we also observe that the Beale-Kato-Majda criterion cannot be improved in the class of $C^{1,alpha}$ solutions.
{"title":"On the stability of self-similar blow-up for $C^{1,alpha}$ solutions to the incompressible Euler equations on $mathbb{R}^3$","authors":"T. Elgindi, T. Ghoul, N. Masmoudi","doi":"10.4310/cjm.2021.v9.n4.a4","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n4.a4","url":null,"abstract":"We study the stability of recently constructed self-similar blow-up solutions to the incompressible Euler equation. A consequence of our work is the existence of finite-energy $C^{1,alpha}$ solutions that become singular in finite time in a locally self-similar manner. As a corollary, we also observe that the Beale-Kato-Majda criterion cannot be improved in the class of $C^{1,alpha}$ solutions.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45197122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-24DOI: 10.4310/cjm.2023.v11.n1.a2
Tanmay Deshpande, S. Mukhopadhyay
In this paper, we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $Gamma$ and a positive integral level $ell$ under the assumption that "$Gamma$ preserves a Borel". As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $Gamma$-crossed modular fusion category as defined by Turaev. To relate these two versions of the Verlinde formula, we formulate the notion of a $Gamma$-crossed modular functor and show that it is very closely related to the notion of a $Gamma$-crossed modular fusion category. We compute the Atiyah algebra and prove (with same assumptions) that the bundles of $Gamma$-twisted conformal blocks associated with a twisted affine Lie algebra define a $Gamma$-crossed modular functor. Along the way, we prove equivalence between a $Gamma$-crossed modular functor and its topological analogue. We then apply these results to derive the Verlinde formula for twisted conformal blocks. We also explicitly describe the crossed S-matrices that appear in the Verlinde formula for twisted conformal blocks.
{"title":"Crossed modular categories and the Verlinde formula for twisted conformal blocks","authors":"Tanmay Deshpande, S. Mukhopadhyay","doi":"10.4310/cjm.2023.v11.n1.a2","DOIUrl":"https://doi.org/10.4310/cjm.2023.v11.n1.a2","url":null,"abstract":"In this paper, we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $Gamma$ and a positive integral level $ell$ under the assumption that \"$Gamma$ preserves a Borel\". As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $Gamma$-crossed modular fusion category as defined by Turaev. To relate these two versions of the Verlinde formula, we formulate the notion of a $Gamma$-crossed modular functor and show that it is very closely related to the notion of a $Gamma$-crossed modular fusion category. We compute the Atiyah algebra and prove (with same assumptions) that the bundles of $Gamma$-twisted conformal blocks associated with a twisted affine Lie algebra define a $Gamma$-crossed modular functor. Along the way, we prove equivalence between a $Gamma$-crossed modular functor and its topological analogue. We then apply these results to derive the Verlinde formula for twisted conformal blocks. We also explicitly describe the crossed S-matrices that appear in the Verlinde formula for twisted conformal blocks.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47972892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-01DOI: 10.4310/cjm.2021.v9.n4.a1
G. Kondyrev, A. Prikhodko
We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck et al.: it relies on the interplay between self dualities of quasiand indcoherent sheaves on X and formal deformation theory of Gaitsgory-Rozenblyum. In particular, we give a description of the Todd class in terms of the difference of two formal group structures on the derived loop scheme LX. The equivariant case is reduced to the non-equivariant one by a variant of the Atiyah-Bott localization theorem.
{"title":"Equivariant Grothendieck–Riemann–Roch theorem via formal deformation theory","authors":"G. Kondyrev, A. Prikhodko","doi":"10.4310/cjm.2021.v9.n4.a1","DOIUrl":"https://doi.org/10.4310/cjm.2021.v9.n4.a1","url":null,"abstract":"We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck et al.: it relies on the interplay between self dualities of quasiand indcoherent sheaves on X and formal deformation theory of Gaitsgory-Rozenblyum. In particular, we give a description of the Todd class in terms of the difference of two formal group structures on the derived loop scheme LX. The equivariant case is reduced to the non-equivariant one by a variant of the Atiyah-Bott localization theorem.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41498499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-02-08DOI: 10.4310/cjm.2020.v8.n3.a1
H. Esnault, M. Kerz
We show that closed subsets of the character variety of a complex variety with negatively weighted homology, which are $p$-adically integral and Galois invariant, are motivic. Final version: Cambridge Journal of Mathematics
{"title":"Arithmetic subspaces of moduli spaces of rank one local systems","authors":"H. Esnault, M. Kerz","doi":"10.4310/cjm.2020.v8.n3.a1","DOIUrl":"https://doi.org/10.4310/cjm.2020.v8.n3.a1","url":null,"abstract":"We show that closed subsets of the character variety of a complex variety with negatively weighted homology, which are $p$-adically integral and Galois invariant, are motivic. Final version: Cambridge Journal of Mathematics","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48501560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-02-07DOI: 10.4310/cjm.2022.v10.n4.a2
J. Bernstein, Lu Wang
For a fixed regular cone in Euclidean space with small entropy we show that all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class.
{"title":"Topological uniqueness for self-expanders of small entropy","authors":"J. Bernstein, Lu Wang","doi":"10.4310/cjm.2022.v10.n4.a2","DOIUrl":"https://doi.org/10.4310/cjm.2022.v10.n4.a2","url":null,"abstract":"For a fixed regular cone in Euclidean space with small entropy we show that all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.6,"publicationDate":"2019-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46770323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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：