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Peking Mathematical Journal
Number of Followers: 0 Hybrid journal (It can contain Open Access articles) ISSN (Print) 2096-6075 - ISSN (Online) 2524-7182 Published by Springer-Verlag [2574 journals] |
- Normal Crossings Degenerations of Symplectic Manifolds
- Abstract: We use local Hamiltonian torus actions to degenerate a symplectic manifold to a normal crossings symplectic variety in a smooth one-parameter family. This construction, motivated in part by the Gross–Siebert and B. Parker’s programs, contains a multifold version of the usual (two-fold) symplectic cut construction and in particular splits a symplectic manifold into several symplectic manifolds containing normal crossings symplectic divisors with shared irreducible components in one step.
PubDate: 2019-09-18
- Abstract: We use local Hamiltonian torus actions to degenerate a symplectic manifold to a normal crossings symplectic variety in a smooth one-parameter family. This construction, motivated in part by the Gross–Siebert and B. Parker’s programs, contains a multifold version of the usual (two-fold) symplectic cut construction and in particular splits a symplectic manifold into several symplectic manifolds containing normal crossings symplectic divisors with shared irreducible components in one step.
- Global Steady Prandtl Expansion over a Moving Boundary III
- Abstract: This is the third paper in a three-part sequence in which we prove that steady, incompressible Navier–Stokes flows posed over the moving boundary, \(y = 0\) , can be decomposed into Euler and Prandtl flows in the inviscid limit globally in \([1, \infty ) \times [0,\infty )\) , assuming a sufficiently small velocity mismatch. In this paper, we prove existence and uniqueness of solutions to the remainder equation.
PubDate: 2019-08-12
- Abstract: This is the third paper in a three-part sequence in which we prove that steady, incompressible Navier–Stokes flows posed over the moving boundary, \(y = 0\) , can be decomposed into Euler and Prandtl flows in the inviscid limit globally in \([1, \infty ) \times [0,\infty )\) , assuming a sufficiently small velocity mismatch. In this paper, we prove existence and uniqueness of solutions to the remainder equation.
- Global Steady Prandtl Expansion over a Moving Boundary II
- Abstract: This is the second paper in a three-part sequence in which we prove that steady, incompressible Navier–Stokes flows posed over the moving boundary, \(y = 0\) , can be decomposed into Euler and Prandtl flows in the inviscid limit globally in \([1, \infty ) \times [0,\infty )\) , assuming a sufficiently small velocity mismatch. In this paper, we develop a functional framework to capture precise decay rates of the remainders, and prove the corresponding embedding theorems by establishing weighted estimates for their higher order tangential derivatives. These tools are then used in conjunction with a third-order energy analysis, which, in particular, enables us to control the nonlinearity \(vu_y\) globally, leading to the main a priori estimate in the analysis.
PubDate: 2019-07-17
- Abstract: This is the second paper in a three-part sequence in which we prove that steady, incompressible Navier–Stokes flows posed over the moving boundary, \(y = 0\) , can be decomposed into Euler and Prandtl flows in the inviscid limit globally in \([1, \infty ) \times [0,\infty )\) , assuming a sufficiently small velocity mismatch. In this paper, we develop a functional framework to capture precise decay rates of the remainders, and prove the corresponding embedding theorems by establishing weighted estimates for their higher order tangential derivatives. These tools are then used in conjunction with a third-order energy analysis, which, in particular, enables us to control the nonlinearity \(vu_y\) globally, leading to the main a priori estimate in the analysis.
- Fully Hodge–Newton Decomposable Shimura Varieties
- Abstract: The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport–Zink spaces and of affine Deligne–Lusztig varieties. We prove a Hodge–Newton decomposition for affine Deligne–Lusztig varieties and for the special fibers of Rapoport–Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge–Newton decomposability) which can be phrased in combinatorial terms. Second, we study the Shimura varieties in which every non-basic \(\sigma \) -isogeny class is Hodge–Newton decomposable. We show that (assuming the axioms of He and Rapoport in Manuscr. Math. 152(3–4):317–343, 2017) this condition is equivalent to nice conditions on either the basic locus or on all the non-basic Newton strata of the Shimura varieties. We also give a complete classification of Shimura varieties satisfying these conditions. While previous results along these lines often have restrictions to hyperspecial (or at least maximal parahoric) level structure, and/or quasi-split underlying group, we handle the cases of arbitrary parahoric level structure and of possibly non-quasi-split underlying groups. This results in a large number of new cases of Shimura varieties where a simple description of the basic locus can be expected. As a striking consequence of the results, we obtain that this property is independent of the parahoric subgroup chosen as level structure. We expect that our conditions are closely related to the question whether the weakly admissible and admissible loci coincide.
PubDate: 2019-06-01
- Abstract: The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport–Zink spaces and of affine Deligne–Lusztig varieties. We prove a Hodge–Newton decomposition for affine Deligne–Lusztig varieties and for the special fibers of Rapoport–Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge–Newton decomposability) which can be phrased in combinatorial terms. Second, we study the Shimura varieties in which every non-basic \(\sigma \) -isogeny class is Hodge–Newton decomposable. We show that (assuming the axioms of He and Rapoport in Manuscr. Math. 152(3–4):317–343, 2017) this condition is equivalent to nice conditions on either the basic locus or on all the non-basic Newton strata of the Shimura varieties. We also give a complete classification of Shimura varieties satisfying these conditions. While previous results along these lines often have restrictions to hyperspecial (or at least maximal parahoric) level structure, and/or quasi-split underlying group, we handle the cases of arbitrary parahoric level structure and of possibly non-quasi-split underlying groups. This results in a large number of new cases of Shimura varieties where a simple description of the basic locus can be expected. As a striking consequence of the results, we obtain that this property is independent of the parahoric subgroup chosen as level structure. We expect that our conditions are closely related to the question whether the weakly admissible and admissible loci coincide.
- Global Steady Prandtl Expansion over a Moving Boundary I
- Abstract: This is the first of three papers in which we prove that steady, incompressible Navier–Stokes flows posed over the moving boundary, \(y = 0\) , can be decomposed into Euler and Prandtl flows in the inviscid limit globally in \([1, \infty ) \times [0,\infty )\) , assuming a sufficiently small velocity mismatch. In this part, sharp decay rates and self-similar asymptotics are extracted for both Prandtl and Eulerian layers.
PubDate: 2019-06-01
- Abstract: This is the first of three papers in which we prove that steady, incompressible Navier–Stokes flows posed over the moving boundary, \(y = 0\) , can be decomposed into Euler and Prandtl flows in the inviscid limit globally in \([1, \infty ) \times [0,\infty )\) , assuming a sufficiently small velocity mismatch. In this part, sharp decay rates and self-similar asymptotics are extracted for both Prandtl and Eulerian layers.
- Holomorphic Anomaly Equations for the Formal Quintic
- Abstract: We define a formal Gromov–Witten theory of the quintic threefold via localization on \({\mathbb {P}}^4\) . Our main result is a direct geometric proof of holomorphic anomaly equations for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov–Witten theory of the quintic threefold. The results suggest that the formal quintic and the true quintic theories should be related by transformations which respect the holomorphic anomaly equations. Such a relationship has been recently found by Q. Chen, S. Guo, F. Janda, and Y. Ruan via the geometry of new moduli spaces.
PubDate: 2019-03-01
- Abstract: We define a formal Gromov–Witten theory of the quintic threefold via localization on \({\mathbb {P}}^4\) . Our main result is a direct geometric proof of holomorphic anomaly equations for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov–Witten theory of the quintic threefold. The results suggest that the formal quintic and the true quintic theories should be related by transformations which respect the holomorphic anomaly equations. Such a relationship has been recently found by Q. Chen, S. Guo, F. Janda, and Y. Ruan via the geometry of new moduli spaces.
- The Fu–Yau Equation in Higher Dimensions
- Abstract: In this paper, we prove the existence of solutions to the Fu–Yau equation on compact Kähler manifolds. As an application, we give a class of non-trivial solutions of the modified Strominger system.
PubDate: 2019-03-01
- Abstract: In this paper, we prove the existence of solutions to the Fu–Yau equation on compact Kähler manifolds. As an application, we give a class of non-trivial solutions of the modified Strominger system.
- Spectrum of SYK Model
- Abstract: This is the first part of a series of papers on the spectrum of the SYK model, which is a simple model of the black hole in physics literature. In this paper, we will give a rigorous proof of the almost sure convergence of the global density of the eigenvalues. We also discuss the largest eigenvalue of the SYK model.
PubDate: 2019-03-01
- Abstract: This is the first part of a series of papers on the spectrum of the SYK model, which is a simple model of the black hole in physics literature. In this paper, we will give a rigorous proof of the almost sure convergence of the global density of the eigenvalues. We also discuss the largest eigenvalue of the SYK model.
- Correction to: Stability of Valuations: Higher Rational Rank
- Abstract: Example 4.2 as well as Lemma 3.31 and Lemma 2.51 have been incorrectly referenced by error. Furthermore, the multi-letter variables such as gr, div, spec have been set in italic instead in roman. The original article has been corrected.
PubDate: 2019-02-01
- Abstract: Example 4.2 as well as Lemma 3.31 and Lemma 2.51 have been incorrectly referenced by error. Furthermore, the multi-letter variables such as gr, div, spec have been set in italic instead in roman. The original article has been corrected.
- On the Isotropic–Nematic Phase Transition for the Liquid Crystal
- Abstract: In this paper, we study the isotropic–nematic phase transition for the nematic liquid crystal based on the Landau–de Gennes \(\mathbf {Q}\) -tensor theory. We justify the limit from the Landau–de Gennes flow to a sharp interface model: in the isotropic region, \(\mathbf {Q}\equiv 0\) ; in the nematic region, the \(\mathbf {Q}\) -tensor is constrained on the manifolds \(\mathcal {N}=\{s_+(\mathbf {n}\otimes \mathbf {n}-\frac{1}{3}\mathbf {I}), \mathbf {n}\in {\mathbb {S}^2}\}\) with \(s_+\) a positive constant, and the evolution of alignment vector field \(\mathbf {n}\) obeys the harmonic map heat flow, while the interface separating the isotropic and nematic regions evolves by the mean curvature flow. This problem can be viewed as a concrete but representative case of the Rubinstein–Sternberg–Keller problem introduced in Rubinstein et al. (SIAM J. Appl. Math. 49:116–133, 1989; SIAM J. Appl. Math. 49:1722–1733, 1989).
PubDate: 2018-12-01
- Abstract: In this paper, we study the isotropic–nematic phase transition for the nematic liquid crystal based on the Landau–de Gennes \(\mathbf {Q}\) -tensor theory. We justify the limit from the Landau–de Gennes flow to a sharp interface model: in the isotropic region, \(\mathbf {Q}\equiv 0\) ; in the nematic region, the \(\mathbf {Q}\) -tensor is constrained on the manifolds \(\mathcal {N}=\{s_+(\mathbf {n}\otimes \mathbf {n}-\frac{1}{3}\mathbf {I}), \mathbf {n}\in {\mathbb {S}^2}\}\) with \(s_+\) a positive constant, and the evolution of alignment vector field \(\mathbf {n}\) obeys the harmonic map heat flow, while the interface separating the isotropic and nematic regions evolves by the mean curvature flow. This problem can be viewed as a concrete but representative case of the Rubinstein–Sternberg–Keller problem introduced in Rubinstein et al. (SIAM J. Appl. Math. 49:116–133, 1989; SIAM J. Appl. Math. 49:1722–1733, 1989).
- Asymptotic of Enumerative Invariants in $${\mathbb {C}}P^2$$ C P 2
- Abstract: In this paper, we give the asymptotic expansion of \(n_{0,d}\) and \(n_{1,d}\) , where \((3d-1+g)!\, n_{g,d}\) counts the number of genus g curves in \({\mathbb {C}}P^2\) through \(3d-1+g\) points in general position and can be identified with certain Gromov–Witten invariants.
PubDate: 2018-12-01
- Abstract: In this paper, we give the asymptotic expansion of \(n_{0,d}\) and \(n_{1,d}\) , where \((3d-1+g)!\, n_{g,d}\) counts the number of genus g curves in \({\mathbb {C}}P^2\) through \(3d-1+g\) points in general position and can be identified with certain Gromov–Witten invariants.
- A Derivation of the Sharp Moserâ€“Trudingerâ€“Onofri Inequalities from the
Fractional Sobolev Inequalities- Abstract: We derive the sharp Moser–Trudinger–Onofri inequalities on the standard n-sphere and CR \((2n+1)\) -sphere as the limit of the sharp fractional Sobolev inequalities for all \(n\ge 1\) . On the 2-sphere and 4-sphere, this was established recently by Chang and Wang. Our proof uses an alternative and elementary argument.
PubDate: 2018-12-01
- Abstract: We derive the sharp Moser–Trudinger–Onofri inequalities on the standard n-sphere and CR \((2n+1)\) -sphere as the limit of the sharp fractional Sobolev inequalities for all \(n\ge 1\) . On the 2-sphere and 4-sphere, this was established recently by Chang and Wang. Our proof uses an alternative and elementary argument.
- Log-Plurigenera in Stable Families of Surfaces
- Abstract: We study the flatness of log-pluricanonical sheaves on stable families of surfaces.
PubDate: 2018-09-01
- Abstract: We study the flatness of log-pluricanonical sheaves on stable families of surfaces.
- Stability of Valuations: Higher Rational Rank
- Abstract: Given a klt singularity \(x\in (X, D)\) , we show that a quasi-monomial valuation v with a finitely generated associated graded ring is a minimizer of the normalized volume function \({\widehat{\text{vol}}}_{(X,D),x}\) , if and only if v induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a minimizer is unique among all quasi-monomial valuations up to rescaling. As a consequence, we prove that for a klt singularity \(x\in X\) on the Gromov–Hausdorff limit of Kähler–Einstein Fano manifolds, the intermediate K-semistable cone associated with its metric tangent cone is uniquely determined by the algebraic structure of \(x\in X\) , hence confirming a conjecture by Donaldson–Sun.
PubDate: 2018-09-01
- Abstract: Given a klt singularity \(x\in (X, D)\) , we show that a quasi-monomial valuation v with a finitely generated associated graded ring is a minimizer of the normalized volume function \({\widehat{\text{vol}}}_{(X,D),x}\) , if and only if v induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a minimizer is unique among all quasi-monomial valuations up to rescaling. As a consequence, we prove that for a klt singularity \(x\in X\) on the Gromov–Hausdorff limit of Kähler–Einstein Fano manifolds, the intermediate K-semistable cone associated with its metric tangent cone is uniquely determined by the algebraic structure of \(x\in X\) , hence confirming a conjecture by Donaldson–Sun.
- Log-Plurigenera in Stable Families
- Abstract: We study the flatness of log-pluricanonical sheaves on stable families of varieties.
PubDate: 2018-09-01
- Abstract: We study the flatness of log-pluricanonical sheaves on stable families of varieties.