• Commun. Anal. Geom. (IF 0.62) Pub Date : 2021-01-08
Ugo Bruzzo; Antonella Grassi

We show that for very general hypersurfaces in odd-dimensional simplicial projective toric varieties satisfying an effective combinatorial property the Hodge conjecture holds. This gives a connection between the Oda conjecture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.

更新日期：2021-01-10
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2021-01-08
F. Reese Harvey; H. Blaine Lawson; Szymon Pliś

This note adapts the sophisticated Richberg technique for approximation in pluripotential theory to the $F$-potential theory associated to a general nonlinear convex subequation $F \subset J^2 (X)$ on a manifold $X$. The main theorem is the following “local to global” result. Suppose $u$ is a continuous strictly $F$-subharmonic function such that each point $x \in X$ has a fundamental neighborhood

更新日期：2021-01-10
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2021-01-08
Chuu-Lian Terng; Zhiwei Wu

A smooth curve $\gamma$ in $\mathbb{R}^{n+1,n}$ is isotropic if $\gamma , \gamma_x, \dotsc , \gamma^{(2n)}_x$ are linearly independent and the span of $\gamma , \gamma_x, \dotsc , \gamma^{(n−1)}_x$ is isotropic. We construct two hierarchies of isotropic curve flows on $\mathbb{R}^{n+1,n}$, whose differential invariants are solutions of Drinfeld–Sokolov’s KdV type soliton hierarchies associated to the

更新日期：2021-01-10
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2021-01-08
Brian Freidin; Yingying Zhang

We study analytic properties of harmonic maps from Riemannian polyhedra into $\operatorname{CAT}(\kappa)$ spaces for $\kappa \in {\lbrace 0, 1 \rbrace}$. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into $\operatorname{CAT}(\kappa)$ spaces. We compute a target variation formula that captures the curvature bound in the target, and use

更新日期：2021-01-10
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2021-01-08
Natasa Sesum; Dong-Ho Tsai; Xiao-Liu Wang

We provide sufficient conditions on an initial curve for the area preserving and the length preserving curvature flows of curves in a plane, to develop a singularity at some finite time or converge to an $m$-fold circle as time goes to infinity. For the area-preserving flow, the positivity of the enclosed algebraic area determines whether the curvature blows up in finite time or not, while for the

更新日期：2021-01-10
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2021-01-08
Steve Bradlow; Lucas Branco; Laura P. Schaposnik

We examine Higgs bundles for non-compact real forms of $SO(4,\mathbb{C})$ and the isogenous complex group $SL(2,\mathbb{C}) \times SL(2,\mathbb{C})$. This involves a study of non-regular fibers in the corresponding Hitchin fibrations and provides interesting examples of non-abelian spectral data.

更新日期：2021-01-10
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2021-01-08
Brian Grajales; Lino Grama; Caio J. C. Negreiros

We describe the invariant metrics on real flag manifolds and classify those with the following property: every geodesic is the orbit of a one-parameter subgroup. Such a metric is called g.o. (geodesic orbit). In contrast to the complex case, on real flag manifolds the isotropy representation can have equivalent submodules, which makes invariant metrics depend on more parameters and allows us to find

更新日期：2021-01-10
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2021-01-08
Brendan Guilfoyle

Counter-examples to the famous conjecture of Carathéodory, as well as the bound on umbilic index proposed by Hamburger, are constructed with respect to Riemannian metrics that are arbitrarily close to the flat metric on Euclidean $3$-space. In particular, Riemannian metrics with a smooth strictly convex $2$-sphere containing a single umbilic point are constructed explicitly, in contradiction with any

更新日期：2021-01-10
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2021-01-08
Gao Chen; Jeff Viaclovsky; Ruobing Zhang

For any elliptic K3 surface $\mathfrak{F} : \mathcal{K} \to \mathbb{P}^1$, we construct a family of collapsing Ricci-flat Kähler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov–Hausdorff limit to $\mathbb{P}^1$ equipped with the McLean metric. There are well-known examples of this type of collapsing, but the key point of our construction is that we can

更新日期：2021-01-10
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-11-01
Po-Ning Chen; Mu-Tao Wang; Shing-Tung Yau

This article considers the quasi-local conserved quantities with respect to a reference spacetime with a cosmological constant. We follow the approach developed by the authors in [7, 26, 27] and define the quasi-local energy as differences of surface Hamiltonians. The ground state for the gravitational energy is taken to be a reference configuration in the de Sitter (dS) or Anti‑de Sitter (AdS) spacetime

更新日期：2020-12-07
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-11-01
David Fajman; Klaus Kröncke

We prove nonlinear stability for a large class of solutions to the Einstein equations with a positive cosmological constant and compact spatial topology in arbitrary dimensions, where the spatial metric is Einstein with either positive or negative Einstein constant. The proof uses the CMC Einstein flow and stability follows by an energy argument.We prove in addition that the development of non-CMC

更新日期：2020-12-07
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-11-01
Abraham Enrique Muñoz Flores; Stefano Nardulli

We show existence and we give a geometric characterization of isoperimetric regions for large volumes, in $C^2$-locally asymptotically Euclidean Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends. This work extends previous results contained in [EM13b], [EM13a], and [BE13]. Moreover strengthening a little bit the speed of convergence to the Schwarzschild metric we

更新日期：2020-12-07
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-11-01
Mehdi Lejmi; Markus Upmeier

The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kähler case. Our main question is the existence of almost Kähler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in

更新日期：2020-12-07
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-11-01
Duncan McCoy

A slope $p/q$ is a characterizing slope for a knot $K$ in $S^3$ if the oriented homeomorphism type of $p/q$-surgery on $K$ determines $K$ uniquely. We show that for each torus knot its set of characterizing slopes contains all but finitely many non-integer slopes. This generalizes work of Ni and Zhang who established such a result for $T_{5,2}$. Along the way we show that if two knots $K$ and $K^\prime$

更新日期：2020-12-07
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-11-01
Miles Simon

We consider solutions $(M, g(t)), 0 \leq t \lt T$, to Ricci flow on compact, connected four dimensional manifolds without boundary. We assume that the scalar curvature is bounded uniformly, and that $T \lt \infty$. In this case, we show that the metric space $(M, d(t))$ associated to $(M, g(t))$ converges uniformly in the $C^0$ sense to $(X, d)$, as $t \nearrow T$, where $(X, d)$ is a $C^0$ Riemannian

更新日期：2020-12-07
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-11-01
Sajjad Lakzian; Christina Sormani

Seven years after the publication of “Smooth convergence away from singular sets” [Communications in Analysis and Geometry 21 (2013), no. 1, 39‑104], Brian Allen discovered a counter example to the published statement of Theorem 1.3. Note that Theorem 4.6 (which is the key theorem cited in other papers) remains correct. We have added an hypothesis to correct the statement of Theorem 1.3 and its consequences

更新日期：2020-12-07
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-12-02
Yi Li; Yuan Yuan; Yuguang Zhang

In this paper, we introduce a geometric flow for Kähler metrics $\omega_t$ coupled with closed $(1,1)$‑forms $\alpha_t$ on a compact Kähler manifold, whose stationary solution is a constant scalar curvature Kähler (cscK) metric, coupled with a harmonic $(1,1)$‑form. We establish the long-time existence, i.e., assuming the initial $(1,1)$‑form $\alpha$ is nonnegative, then the flow exists as long as

更新日期：2020-12-03
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-12-02
Chang-Shou Lin; Chin-Lung Wang

Let $E = \mathbb{C} / \Lambda$ be a flat torus and $G$ be its Green function with singularity at $0$. Consider the multiple Green function $G_n$ on $E^n$:$G_n (z_1, \dotsc , z_n) := \sum_{i \lt j} G (z_i - z_j) - n \sum^n_{i=1} G (z_i) \: \textrm{.}$A critical point $a = (a_1, \dotsc , a_n)$ of $G_n$ is called trivial if $\lbrace a_1, \dotsc , a_n \rbrace = \lbrace -a_1, \dotsc , -a_n \rbrace$. For

更新日期：2020-12-03
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-12-02
Longzhi Lin

In the last 15 years, the series of works of White and Huisken–Sinestrari yield that the blowup limits at singularities are convex for the mean curvature flow of mean convex hypersurfaces. In 1998 Smoczyk [20] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in $\mathbf{R}^3$.We prove in this paper that this

更新日期：2020-12-03
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-12-02
Alexis Michelat

We obtain an upper bound for the Morse index of Willmore spheres $\Sigma \subset S^3$ coming from an immersion of $S^2$. The quantization of Willmore energy, which is a consequence of the classification of Willmore spheres in $S^3$ by Robert Bryant, shows that there exists an integer $m$ such that $\mathscr{W} (\Sigma) = 4 \pi m$. We show that the Morse index $\operatorname{Ind}_\mathscr{W} (\Sigma)$

更新日期：2020-12-03
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-12-02
Mason Pember

We give a detailed account of the gauge-theoretic approach to Lie applicable surfaces and the resulting transformation theory. In particular, we show that this approach coincides with the classical notion of $\Omega$‑ and $\Omega$‑surfaces of Demoulin.

更新日期：2020-12-03
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-12-02
Naoki Shioji; Kohtaro Watanabe

We consider a variational problem of $p$-elastic curves in two-dimensional sphere. We give its first variation formula, and in two-dimensional sphere, we give a realization of a solution which satisfies that the first variation formula is zero. We also show the existence of a flat-core, closed $p$-elastic curve.

更新日期：2020-12-03
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-09-01
Spiro Karigiannis; Jason D. Lotay

We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) $\mathrm{G}_2$ manifolds. In the AC case, we show that if the rate of convergence ν to the cone at infinity is generic in a precise sense and lies in the interval $(-4, 0)$, then the moduli space is smooth and we compute its dimension in terms of topological and analytic data. For generic rates $\nu \lt 更新日期：2020-10-15 • Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-09-01 Francesco Lin We refine some classical estimates in Seiberg–Witten theory, and discuss an application to the spectral geometry of three-manifolds. We show that for any Riemannian metric on a rational homology three-sphere$Y$, the first eigenvalue of the Hodge Laplacian on coexact one-forms is bounded above explicitly in terms of the Ricci curvature, provided that$Y$is not an$L$-space (in the sense of Floer homology) 更新日期：2020-10-15 • Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-09-01 Peter M. Topping; Hao Yin We prove that the Ricci flow that contracts a hyperbolic cusp has curvature decay$\operatorname{max} K \sim \frac{1}{t^2}$. In order to do this, we prove a new Li–Yau type differential Harnack inequality for Ricci flow. 更新日期：2020-10-15 • Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-07-01 Christopher J. Bishop; Claude Lebrun It is known that the almost-Kähler anti-self-dual metrics on a given$4$-manifold sweep out an open subset in the moduli space of antiself- dual metrics. However, we show here by example that this subset is not generally closed, and so need not sweep out entire connected components in the moduli space. Our construction hinges on an unexpected link between harmonic functions on certain hyperbolic$3$-manifolds 更新日期：2020-07-01 • Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-07-01 Christine Breiner; Ailana Fraser; Lan-Hsuan Huang; Chikako Mese; Pam Sargent; Yingying Zhang Let$\varphi \in C^0 \cap W_{1,2} (\Sigma, X)$where$\Sigma$is a compact Riemann surface,$X$is a compact locally CAT(1) space, and$W_{1,2} (\Sigma, X)$is defined as in Korevaar–Schoen. We use the technique of harmonic replacement to prove that either there exists a harmonic map$u : \Sigma \to X$homotopic to$\varphi$or there exists a nontrivial conformal harmonic map$v : \mathbb{S}^2 \to

更新日期：2020-07-01
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-07-01
Frances Kirwan

Let $K$ be a compact Lie group and fix an invariant inner product on its Lie algebra $\mathfrak{k}$. Given a Hamiltonian action of $K$ on a compact symplectic manifold $X$ with moment map $\mu : X \to \mathfrak{k}^\ast$, the normsquare ${\lVert \mu \rVert}^2$ of $\mu$ defines a Morse stratification $\lbrace S_\beta : \beta \in \mathcal{B} \rbrace$ of $X$ by locally closed symplectic submanifolds of

更新日期：2020-07-01
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-07-01
Rafe Mazzeo; Edward Witten

It is conjectured that the coefficients of the Jones polynomial can be computed by counting solutions of the KW equations on a fourdimensional half-space, with certain boundary conditions that depend on a knot. The boundary conditions are defined by a “Nahm pole” away from the knot with a further singularity along the knot. In a previous paper, we gave a precise formulation of the Nahm pole boundary

更新日期：2020-07-01
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-07-01
Sagun Chanillo; Magdalena Czubak; Dana Mendelso; Andrea Nahmod; Gigliola Staffilani

We study nonlinear wave equations on $\mathbb{R}^{2+1}$ with quadratic derivative nonlinearities, which include in particular nonlinearities exhibiting a null form structure, with random initial data in $H^1_x \times L^2_x$. In contrast to the counterexamples of Zhou [73] and Foschi–Klainerman [23], we obtain a uniform time interval $I$ on which the Picard iterates of all orders are almost surely bounded

更新日期：2020-07-01
• Commun. Anal. Geom. (IF 0.62) Pub Date : 2020-07-01
Antonella Marini; Rachel Maitra; Vincent Moncrief

In this article we discuss our ongoing program to extend the scope of certain, well-developed microlocal methods for the asymptotic solution of Schrödinger’s equation (for suitable ‘nonlinear oscillatory’ quantum mechanical systems) to the treatment of several physically significant, interacting quantum field theories. Our main focus is on applying these ‘Euclidean-signature semi-classical’ methods

更新日期：2020-07-01
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