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  • Deformation theory of $\mathrm{G}_2$ conifolds
    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
  • Monopole Floer homology and the spectral geometry of three-manifolds
    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
  • Rate of curvature decay for the contracting cusp Ricci flow
    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
  • Bipyramid decompositions of multicrossing link complements
    Commun. Anal. Geom. (IF 0.62) Pub Date : 
    Colin Adams; Gregory Kehne

    Generalizing previous constructions, we present a dual pair of decompositions of the complement of a link $L$ into bipyramids, given any multicrossing projection of $L$. When $L$ is hyperbolic, this gives new upper bounds on the volume of $L$ given its multicrossing projection. These bounds are realized by three closely related infinite tiling weaves.

    更新日期:2020-07-20
  • Gronwall’s conjecture for $3$-webs with infinitesimal symmetries
    Commun. Anal. Geom. (IF 0.62) Pub Date : 
    Sergey I. Agafonov

    We study non-flat planar 3‑webs with infinitesimal symmetries. Using multi-dimensional Schwarzian derivative we give a criterion for linearization of such webs and present a projective classification thereof. Using this classification we show that the Gronwall conjecture is true for 3‑webs admitting infinitesimal symmetries.

    更新日期:2020-07-20
  • Remarks on complete noncompact Einstein warped products
    Commun. Anal. Geom. (IF 0.62) Pub Date : 
    R. Batista; M. Ranieri; E. Ribeiro

    The purpose of this article is to investigate the structure of complete non-compact quasi-Einstein manifolds. We show that complete noncompact quasi-Einstein manifolds with $\lambda = 0$ are connected at infinity. In addition, we provide some conditions under which quasi-Einstein manifolds with $\lambda \lt 0$ are $f$-non-parabolic. In particular, we obtain estimates on volume growth of geodesic balls

    更新日期:2020-07-20
  • On Li–Yau gradient estimate for sum of squares of vector fields up to higher step
    Commun. Anal. Geom. (IF 0.62) Pub Date : 
    Der-Chen Chang; Shu-Cheng Chang; Chien Lin

    In this paper, we generalize Cao–Yau’s gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li–Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As

    更新日期:2020-07-20
  • From Lagrangian to totally real geometry: coupled flows and calibrations
    Commun. Anal. Geom. (IF 0.62) Pub Date : 
    Jason D. Lotay; Tommaso Pacini

    We show that the properties of Lagrangian mean curvature flow are a special case of a more general phenomenon, concerning couplings between geometric flows of the ambient space and of totally real submanifolds. Both flows are driven by ambient Ricci curvature or, in the non-Kähler case, by its analogues. To this end we explore the geometry of totally real submanifolds, defining (i) a new geometric

    更新日期:2020-07-20
  • Existence results for some problems on Riemannian manifolds
    Commun. Anal. Geom. (IF 0.62) Pub Date : 
    Giovanni Molica Bisci; Dušan Repovš; Luca Vilasi

    By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional $(d \geq 3)$ Riemannian manifold without boundary. As a direct consequence of our main theorems, we prove the existence of at least one solution to the following Yamabe-type problem\[\begin{cases}-\Delta_g w + \alpha(\sigma) w = \mu K(\sigma) w^{\frac{d+2}{d-2}}

    更新日期:2020-07-20
  • Some integral curvature estimates for the Ricci flow in four dimensions
    Commun. Anal. Geom. (IF 0.62) Pub Date : 
    Miles Simon

    We consider solutions $(M^4 , g(t)), 0 \leq t \lt T$, to Ricci flow on compact, four-dimensional manifolds without boundary. We prove integral curvature estimates which are valid for any such solution. In the case that the scalar curvature is bounded and $T \lt \infty$, we show that these estimates imply that the (spatial) integral of the square of the norm of the Riemannian curvature is bounded by

    更新日期:2020-07-20
  • A gap theorem of four-dimensional gradient shrinking solitons
    Commun. Anal. Geom. (IF 0.62) Pub Date : 
    Zhuhong Zhang

    In this paper, we will prove a gap theorem on four-dimensional gradient shrinking soliton. More precisely, we will show that any complete four-dimensional gradient shrinking soliton with nonnegative and bounded Ricci curvature, satisfying a pinched Weyl curvature, either is flat, or $\lambda_1 + \lambda_2 \geq c_0 R \gt 0$ at all points, where $c_0 \approx 0.29167$ and $\lbrace \lambda_i \rbrace$ are

    更新日期:2020-07-20
  • Anti-self-dual $4$-manifolds, quasi-Fuchsian groups, and almost-Kähler geometry
    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
  • Existence of harmonic maps into CAT(1) spaces
    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
  • Symplectic quotients of unstable Morse strata for normsquares of moment maps
    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
  • The KW equations and the Nahm pole boundary condition with knots
    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
  • Almost sure boundedness of iterates for derivative nonlinear wave equations
    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
  • A Euclidean signature semi-classical program
    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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