We introduce a local-in-time existence and uniqueness class for solutions to the 2d Euler equation with unbounded vorticity. Furthermore, we show that solutions belonging to this class can develop stronger singularities in finite time, meaning that they experience finite time blow-up and exit the well-posedness class. Such solutions may be continued as weak solutions (potentially non-uniquely) after the singularity. While the general dynamics of 2d Euler solutions beyond the Yudovich class will certainly not be so tame, studying such solutions gives a way to study singular phenomena in a more controlled setting.
We establish a criterion for determining when a smooth Deligne-Mumford stack is a weighted blow-up. More precisely, given a smooth Deligne-Mumford stack X and a Cartier divisor E subset of X such that (1) E is a weighted projective bundle over a smooth Deligne-Mumford stackYand (2) for every y is an element of Y we have O-X(E)|(Ey)}}similar to q O-Ey}}(-1), there exists a contraction X oZ to a smooth Deligne-Mumford stack Z. Moreover, the stack X can be recovered as a weighted blow-up along Y subset of Z with exceptional divisor E, and Z is a push-out in the category of algebraic stacks. As an application, we show that the moduli stack (M)over bar(1,n) of stable n-pointed genus 1 curves is a weighted blow-up of the stack of pseudo-stable curves. A key step is a reconstruction result for smooth Deligne-Mumford stacks that may be of independent interest.
We show that every element of \mathrm{SL}_{n}(\mathbb{Z}/q\mathbb{Z}) can be lifted to an element of \mathrm{SL}_{n}(\mathbb{Z}) of norm at most Cq^{2}\log q , while there exists an element such that every lift of it is of norm at least q^{2+o(1)} . This should be compared to the recent result that almost every element has a lift of norm bounded by q^{1+1/n+o(1)} . The main step in the proof is showing that for every q , there is a small element in (\mathbb{Z}/q\mathbb{Z})^{\times} with a large n -th root, which is a result of independent interest. In the proof we use tools from additive combinatorics including Bohr sets.
We study a good shape property of boundary sections of convex solutions to the oblique boundary value problem for the Monge–Ampère equation \det D^{2}u =f(x) \quad \text{in }\Omega, \quad D_{\beta}u = \phi(x) \quad \text{on }\partial \Omega. In two dimensions, we prove a global C^{2,\alpha} estimate for solutions. For dimensions n \geq 3 , we show that this estimate remains valid provided the solution satisfies a quadratic growth condition in tangential directions. We also prove an existence result for convex solutions to the Monge–Ampère equation with an oblique Robin boundary condition.
Theorem 2.8 from [Free Banach lattices, J. Eur. Math. Soc. (2024), online first] regarding the order density of FVL [E] in FBL^{(p)}[E] only holds when the underlying Banach space E is finite dimensional. We correct this issue and re-evaluate the minor consequences on the paper.
Let \Gamma be a group acting with finite stabilizers and finite fundamental domain on a building of type \widetilde A_{2} . We prove that any non-trivial normal subgroup of \Gamma is of finite index in \Gamma .
We correct several issues in our paper [J. Eur. Math. Soc. 24, 3873–3949 (2022)], detected by Larson, Payne, and Stapledon. The results of the paper are not affected by the corrections, but one update may be of interest for the future research: it is a previously missed case in the classification of 3-dimensional B -facets (Lemma 5.18). It corrects expectations on how this classification might look in higher dimensions.
We study homological mirror symmetry for (\mathbb{P}^{2}, \Omega) viewed as an object of birational geometry, with \Omega the standard meromorphic volume form. First, we construct universal objects on the two sides of mirror symmetry, focusing on the exact symplectic setting: a smooth complex scheme U_{\mathrm{univ}} and a Weinstein manifold M_{\mathrm{univ}} , both of infinite type. We prove homological mirror symmetry for them. Second, we consider autoequivalences. We prove that automorphisms of U_{\mathrm{univ}} are given by a natural discrete subgroup of \mathrm{Bir}(\mathbb{P}^{2}, \pm \Omega) , and that all of these automorphisms are mirror to symplectomorphisms of M_{\mathrm{univ}} . We conclude with some applications.
We show that every element of SLn(Z/qZ) can be lifted to an element of SLn(Z) of norm at most Cq(2)log q, while there exists an element such that every lift of it is of norm at least q(2 + o(1)). This should be compared to the recent result that almost every element has a lift of norm bounded by q(1+1/n+o(1)). The main step in the proof is showing that for every q, there is a small element in (Z/qZ) (x) with a large n-th root, which is a result of independent interest. In the proof we use tools from additive combinatorics including Bohr sets.
By using several new crucial a priori estimates, we provide a comprehensive resolution of first order generic mean field type control problems and also establish the global-in-time existence and uniqueness of classical solutions of their Bellman and master equations. Rather than developing the analytical approach via tackling the Bellman and master equation directly, we apply the maximum principle approach by considering the induced forward-backward ordinary differential equation (FBODE) system; indeed, we first show the local-in-time unique existence of the solution of the FBODE system for a variety of terminal data by a Banach fixed point argument, and then provide crucial a priori estimates bounding the derivatives of the decoupling field of FBODE by utilizing a monotonicity condition that can be deduced from the positive definiteness of the Schur complement of the Hessian matrix of the Lagrangian in the lifted version and manipulating the first order condition appropriately; this uniform bound over the whole planning horizon [0,T] allows us to partition [0,T] into a finite number of subintervals with a common small length and then glue the consecutive local-in-time solutions together to form the unique global-in-time solution of the FBODE system. The regularity of the global-in-time solution follows from that of the local ones due to the regularity assumptions on the coefficient functions. Moreover, the regularity of the value function will also be shown with the aid of the regularity of the solution couple of the FBODE system and the regularity assumptions on the coefficient functions, with which we can further deduce that this value function and its linear functional derivative satisfy the Bellman and master equations, respectively; further analysis of the unique nature of the FBODE solution implies the uniqueness of the classical solutions of those equations. Finally, to illustrate the effectiveness of our proposed general theory, we also provide the resolution of some nontrivial non-linear-quadratic examples with nonseparable Hamiltonian which have not yet been handled in the literature.
We consider the eigenvalue problem A(S)2 xi+ 2 xi= 0 in Q and = 0 along partial derivative Omega, where Omega is the complement of a finite disjoint union of smooth, bounded, simply connected regions in the two-sphere S-2. Assuming that |del xi| is locally constant along partial derivative Omega and that xi has infinitely many maximum points, we classify positive solutions: every positive solution is rotationally symmetric. As a consequence, we obtain a characterization of the critical catenoid as the only embedded free boundary minimal annulus in the unit ball whose support function has infinitely many critical points.
The real locus of the moduli space of stable genus zero curves with marked points, M-0,n+1(-)(R), is known to be a smooth manifold and is the Eilenberg-MacLane spaces for the so-called pure cactus groups. We describe the operad formed by these spaces in terms of a homotopy quotient of an operad of associative algebras. Using this model, we identify various Hopf models for the algebraic operad of chains and homologies of M-0;n+1(-)(R). In particular, we show that the operad M-0,n+1(-)(R) is not formal. As an application of these operadic constructions, we prove that for each n, the cohomology ring (HM0;n+1-)-M-center dot(R);Q) is a Koszul algebra, and that the manifold M-0;n+1(-)(R) is not formal for n >= 6 but is a rational K(pi,1)-space. Additionally, we describe the Lie algebras associated with the lower central series filtration of the pure cactus groups.
We study a good shape property of boundary sections of convex solutions to the oblique boundary value problem for the Monge-Amp & egrave;re equation det D(2)u D f (x) in Omega, D(beta)u=Phi(x) on @Omega. In two dimensions, we prove a global C-2,C-alpha estimate for solutions. For dimensions n >= 3, we show that this estimate remains valid provided the solution satisfies a quadratic growth condition in tangential directions. We also prove an existence result for convex solutions to the Monge-Ampre equation with an oblique Robin boundary condition.
We consider the problem of the strong density of smooth maps in the Sobolev space W-s,W-p (Q(m); N), where 0 < s < +infinity, 1 <= p < +infinity, Q(m) is the unit cube in R-m, and N is a smooth compact connected Riemannian manifold without boundary. Our main result fully answers the strong density problem in the whole range 0 < s < +infinity: the space C-infinity(Q(m); N) is dense in W-s,W-p (Q(m);N) if and only if pi([sp])(N) = {0}. This completes the results of Bethuel (s = 1), Brezis and Mironescu (0 < s < 1), and Bousquet, Ponce, and Van Schaftingen (s = 2, 3, & mldr;). We also consider the case of more general domains Omega, in the setting studied by Hang and Lin when s = 1.
We establish that frame flows for geometrically finite hyperbolic manifolds of arbitrary dimensions Gamma\Hd+1 are exponentially mixing with respect to the Bowen-Margulis-Sullivan measure, which is the measure of maximal entropy. This paper focuses on the remaining case, the case with cusps. To prove this, we utilize the countably infinite symbolic coding and perform a frame flow version of Dolgopyat's method & agrave; la Sarkar-Winter and Tsujii-Zhang. This requires the local non-integrability condition and the non-concentration property but the challenge in the presence of cusps is that the latter holds only on a large proper subset. To overcome this, we use an effective renewal theorem to prove a uniform large deviation property for symbolic recurrence to the large subset, inspired by the work of Li. Applications of the main theorem include an asymptotic formula for matrix coefficients for L-2 .(Gamma\SO.(d + 1)(omicron) )with an exponential error term, and exponential equidistribution of holonomies and translates of horospherical orbits
We study homological mirror symmetry for (P-2, Omega) viewed as an object of birational geometry, with Omega the standard meromorphic volume form. First, we construct universal objects on the two sides of mirror symmetry, focusing on the exact symplectic setting: a smooth complex scheme U-univ and a Weinstein manifold M-univ, both of infinite type. We prove homological mirror symmetry for them. Second, we consider autoequivalences. We prove that automorphisms of Uuniv are given by a natural discrete subgroup of Bir(P-2, +/-Omega), and that all of these automorphisms are mirror to symplectomorphisms of M-univ. We conclude with some applications.
If A is a set of natural numbers of exponential density 8, then the exponential density of all numbers of the form x(3) + a with x is an element of N and alpha is an element of A is at least min(1, (1)/3 + 5/(6)delta). This is a considerable improvement on the previous best lower bounds for this problem, obtained by Davenport more than 80 years ago. The result is the best possible for delta > (4) /5 .
For closed connected Riemannian spin manifolds, an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number of geometric consequences. Some are known and include Llarull's scalar curvature rigidity of the standard metric on the sphere, Geroch's conjecture on the impossibility of positive scalar curvature on tori, and a mean curvature estimate for spin fill-ins with nonnegative scalar curvature due to Gromov, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler. New applications provide a comparison of the hyperspherical radius with the Yamabe constant and improved estimates of the hyperspherical radius for K & auml;hler manifolds, K & auml;hler-Einstein manifolds, quaternionic K & auml;hler manifolds, and manifolds with a harmonic 1-form of constant length.
The biharmonic flow of hypersurfaces M-n immersed in the Euclidean space Rn+1 for n >= 2 is given by a fourth order geometric evolution equation, which is similar to the Willmore flow. We apply the Michael-Simon Sobolev inequality to establish new Gagliardo-Nirenberg inequalities on hypersurfaces. Based on these Gagliardo-Nirenberg inequalities, we apply local energy estimates to extend the solution by a covering argument and obtain an estimate on the maximal existence time of the biharmonic flow of hypersurfaces in higher dimensions. In particular, we solve a problem posed by Bernard et al. (2019) on the biharmonic hypersurface flow for n = 4. Finally, we apply our new approach to prove global existence of the Willmore flow in higher dimensions
