This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print: the exceptions are (1) the statement that manifolds that can collapse with local lower bound on sectional curvature are graph manifolds - this is deferred to a separate paper, since the proof has nothing to do with the Ricci flow, and (2) the claim on the lower bound for the volume of maximal horns and the smoothness of solutions from some time on, which turned out to be unjustified and, on the other hand, irrelevant for the other conclusions.
Let M be a closed oriented three-manifold, whose prime decomposition contains no aspherical factors. We show that for any initial riemannian metric on M the solution to the Ricci flow with surgery, defined in our previous paper math.DG/0303109, becomes extinct in finite time. The proof uses a version of the minimal disk argument from 1999 paper by Richard Hamilton, and a regularization of the curve shortening flow, worked out by Altschuler and Grayson.
AbstractWe extend the Auslander–Iyama correspondence to the setting of exact dg categories. By specializing it to exact dg categories concentrated in degree zero, we obtain a generalization of the higher Auslander correspondence for exact categories due to Ebrahimi–Nasr-Isfahani (in the case of exact categories with split retractions).
In arXiv:2207.08617 [math.DG] Brendle-Hirsch-Johne proved that $T^m\times S^{n-m}$ does not admit metrics with positive $m$-intermediate curvature when $n\leq 7$. Chu-Kwong-Lee showed in arXiv:2208.12240 [math.DG] a corresponding rigidity statement when $n\leq 5$. In this paper, we show the sharpness of the dimension constraints by giving concrete counterexamples in $n\geq 7$ and extending the rigidity result to $n=6$. Concerning uniformly positive intermediate curvature, we show that simply-connected manifolds with dimension $\leq 5$ and bi-Ricci curvature $\geq 1$ have finite Urysohn 1-width. Counterexamples are constructed in dimension $\geq 6$.
Under natural conditions, the null distance introduced by Sormani and Vega (2016 Class. Quantum Grav. 33 085001) is a metric space distance function on spacetime, which, in a certain precise sense, can encode the causality of spacetime. The null distance function requires the choice of a time function. The purpose of this note is to observe that the causality assumptions related to such a choice in results used to establish global encoding of causality, due to Sakovich and Sormani (2023 J. Math. Phys. 64 012502) and to Burtscher and García-Heveling (2022 arXiv:2209.15610 [math.DG]), can be weakened.
Aulacaspis yasumatsui Takagi invaded Guam in 2003 and caused the widespread mortality of the indigenous Cycas micronesica K.D. Hill population. The regeneration of the surviving tree population continues to be constrained 20 years later, and a look at the changes in megastrobili traits may inform future conservation management decisions concerning regeneration. We quantified megastrobilus reproductive effort and output from 2001 to 2022 to address this need. The reproductive effort of each megastrobilus was immediately reduced by the invasion, as the number of megasporophylls declined by 29%, and the number of ovules declined by 73% in 2006. Reproductive output was also damaged, as the percent seed set declined by 56% and the number of seeds per strobilus declined by 88%. These fecundity metrics have shown few signs of recovery through 2022. Our results reveal that chronic A. yasumatsui infestations, combined with other invasive herbivore threats, have damaged the host C. micronesica population through a sustained reduction in ovule production and the percent seed set for each megastrobilus, thereby impairing regeneration. This plant response to the biotic threats is distinct from the ongoing mortality of mature trees and emerging seedlings. Conservation interventions may be required to foster a return to adequate regeneration during future attempts to aid C. micronesica recovery.
Local foliations of area constrained Willmore surfaces on a 3-dimensional Riemannian manifold were constructed by Lamm et al (2020 Ann. Inst. Fourier 70 1639–62) and Ikoma et al (2020 Int. Math. Res. Not. 70 6538–68), the leaves of these foliations are in particular critical surfaces of the Hawking energy in case they are contained in a totally geodesic spacelike hypersurface. We generalize these foliations to the general case of a non-totally geodesic spacelike hypersurface, constructing an unique local foliation of area constrained critical surfaces of the Hawking energy. A discrepancy when evaluating the so called small sphere limit of the Hawking energy was found by Friedrich (2020 arXiv:1909.02388v2 [math.DG]), he studied concentrations of area constrained critical surfaces of the Hawking energy and obtained a result that apparently differs from the well established small sphere limit of the Hawking energy of Horowitz and Schmidt (1982 Proc. R. Soc. A 381 215–24), this small sphere limit in principle must be satisfied by any quasi local energy. We independently confirm the discrepancy and explain the reasons for it to happen. We also prove that these surfaces are suitable to evaluate the Hawking energy in the sense of Lamm et al (2011 Math. Ann. 350 1–78), and we find an indication that these surfaces may induce an excess in the energy measured.
In this article, we study eternal solutions to the Allen-Cahn equation in the 3-sphere, in view of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral Brakke flows that connect Clifford tori to equatorial spheres, and study a family of such flows, in particular their symmetry properties. Our approach is based on the realization of Brakke’s motion by mean curvature as a singular limit of Allen-Cahn gradient flows, as studied by Ilmanen (J Differ Geom 38(2):417–461, 1993) and Tonegawa (Hiroshima Math J 33(3): 323–341, 2003), and it uses the classification of ancient gradient flows in spheres, by Choi and Mantoulidis (Amer J Math, 2019), as well as the rigidity of stationary solutions with low Morse index proved by Hiesmayr (arXiv:2007.08701 [math.DG], 2020).
Min–max theory for the Allen–Cahn equation was developed by Guaraco (J Differ Geom 108:91–133, 2018) and Gaspar–Guaraco (Calc Var Partial Differ Equ 57:101, 42, 2018). They showed that the Allen–Cahn widths are greater than or equal to the Almgren–Pitts widths. In this article we will prove that the reverse inequalities also hold, i.e. the Allen–Cahn widths are less than or equal to the Almgren–Pitts widths. Hence, the Almgren–Pitts widths and the Allen–Cahn widths coincide. We will also show that all the closed minimal hypersurfaces (with optimal regularity), which are obtained from the Allen–Cahn min–max theory, are also produced by the Almgren–Pitts min–max theory. As a consequence, we will point out that the index upper bound in the Almgren–Pitts setting, proved by Marques–Neves (Camb J Math 4(4):463–511, 2016) and Li (An improved Morse index bound of min–max minimal hypersurfaces, 2020. arXiv:2007.14506 [math.DG]), can also be obtained from the index upper bound in the Allen–Cahn setting, proved by Gaspar (J Geom Anal 30:69-85, 2020) and Hiesmayr (Commun Partial Differ Equ 43(11):1541–1565, 2018).
In this article, we study the second variation of the energy functional associated to the Allen–Cahn equation on closed manifolds. Extending well-known analogies between the gradient theory of phase transitions and the theory of minimal hypersurfaces, we prove the upper semicontinuity of the eigenvalues of the stability operator and consequently obtain upper bounds for the Morse index of limit interfaces which arise from solutions with bounded energy and index without assuming any multiplicity or orientability condition on these hypersurfaces. This extends some recent results of Le (Indiana Univ Math J 60:1843–1856, 2011; J Math Pures Appl 103:1317–1345, 2015)) and Hiesmayr (arXiv:1704.07738 preprint [math.DG], 2017).
Let $M$ be a Calabi-Yau $m$-fold, and consider compact, graded Lagrangians $L$ in $M$. Thomas and Yau math.DG/0104196, math.DG/0104197 conjectured that there should be a notion of "stability" for such $L$, and that if $L$ is stable then Lagrangian mean curvature flow $\{L^t:t\in[0,\infty)\}$ with $L^0=L$ should exist for all time, and $L^\infty=\lim_{t\to\infty}L^t$ should be the unique special Lagrangian in the Hamiltonian isotopy class of $L$. This paper is an attempt to update the Thomas-Yau conjectures, and discuss related issues. It is a folklore conjecture that there exists a Bridgeland stability condition $(Z,\mathcal P)$ on the derived Fukaya category $D^b\mathcal F(M)$ of $M$, such that an isomorphism class in $D^b\mathcal F(M)$ is $(Z,\mathcal P)$-semistable if (and possibly only if) it contains a special Lagrangian, which must then be unique. We conjecture that if $(L,E,b)$ is an object in an enlarged version of $D^b\mathcal F(M)$, where $L$ is a compact, graded Lagrangian in $M$ (possibly immersed, or with "stable singularities"), $E\to M$ a rank one local system, and $b$ a bounding cochain for $(L,E)$ in Lagrangian Floer cohomology, then there is a unique family $\{(L^t,E^t,b^t):t\in[0,\infty)\}$ such that $(L^0,E^0,b^0)=(L,E,b)$, and $(L^t,E^t,b^t)\cong(L,E,b)$ in $D^b\mathcal F(M)$ for all $t$, and $\{L^t:t\in[0,\infty)\}$ satisfies Lagrangian MCF with surgeries at singular times $T_1,T_2,\dots,$ and in graded Lagrangian integral currents we have $\lim_{t\to\infty}L^t=L_1+\cdots+L_n$, where $L_j$ is a special Lagrangian integral current of phase $e^{i\pi\phi_j}$ for $\phi_1>\cdots>\phi_n$, and $(L_1,\phi_1),\ldots,(L_n,\phi_n)$ correspond to the decomposition of $(L,E,b)$ into $(Z,\mathcal P)$-semistable objects. We also give detailed conjectures on the nature of the singularities of Lagrangian MCF that occur at the finite singular times $T_1,T_2,\ldots.$
The main purpose of this article is to classify contact structures on some 3-manifolds, namely lens spaces, most torus bundles over a circle, the solid torus, and the thickened torus T^2 x [0,1]. This classification completes earlier work (by Etnyre [math.DG/9812065], Eliashberg, Kanda, Makar-Limanov, and the author) and results from the combination of two techniques: surgery, which produces many contact structures, and tomography, which allows one to analyse a contact structure given a priori and to create from it a combinatorial image. The surgery methods are based on a theorem of Y. Eliashberg -- revisited by R. Gompf [math.GT/9803019] -- and produces holomorphically fillable contact structures on closed manifolds. Tomography theory, developed in parts 2 and 3, draws on notions introduced by the author and yields a small number of possible models for contact structures on each of the manifolds listed above.