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Differential Harnack inequalities via Concavity of the arrival time Commun. Anal. Geom. (IF 0.7) Pub Date : 2024-01-04 Theodora Bourni, Mat Langford
We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set flow equations. In particular, this yields a short proof of Hamilton’s differential
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Embedded totally geodesic surfaces in fully augmented links Commun. Anal. Geom. (IF 0.7) Pub Date : 2024-01-04 Sierra Knavel, Rolland Trapp
This paper studies embedded totally geodesic surfaces in fully augmented link complements. Not surprisingly, there are no closed embedded totally geodesic surfaces. Non-compact surfaces disjoint from crossing disks are seen to be punctured spheres orthogonal to the standard cell decomposition, while those that intersect crossing disks do so in very restricted ways. Finally we show there is an augmentation
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Sharp entropy bounds for plane curves and dynamics of the curve shortening flow Commun. Anal. Geom. (IF 0.7) Pub Date : 2024-01-04 Julius Baldauf, Ao Sun
We prove that a closed immersed plane curve with total curvature $2 \pi m$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2 \pi m$ whose entropy is less than $m$ times the entropy of the embedded circle. As an application, we extend Colding–Minicozzi’s
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Cohomogeneity one Ricci solitons from Hopf fibrations Commun. Anal. Geom. (IF 0.7) Pub Date : 2024-01-04 Matthias Wink
This paper studies cohomogeneity one Ricci solitons. If the isotropy representation of the principal orbit $G/K$ consists of two inequivalent $\operatorname{Ad}_K$-invariant irreducible summands, the existence of continuous families of non-homothetic complete steady and expanding Ricci solitons on non-trivial bundles is shown. These examples were detected numerically by Buzano–Dancer–Gallaugher–Wang
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Deformation theory of nearly $G_2$ manifolds Commun. Anal. Geom. (IF 0.7) Pub Date : 2024-01-04 Shubham Dwivedi, Ragini Singhal
$\def\G{\mathrm{G}_2}$We study the deformation theory of nearly $\G$ manifolds. These are seven-dimensional manifolds admitting real Killing spinors. We show that the infinitesimal deformations of nearly $\G$ structures are obstructed in general. Explicitly, we prove that the infinitesimal deformations of the homogeneous nearly $\G$ structure on the Aloff–Wallach space are all obstructed to second
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Constant mean curvature $n$-noids in hyperbolic space Commun. Anal. Geom. (IF 0.7) Pub Date : 2024-01-04 Thomas Raujouan
Using the DPW method, we construct genus zero Alexandrov-embedded constant mean curvature (greater than one) surfaces with any number of Delaunay ends in the hyperbolic space.
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Vanishing time behavior of solutions to the fast diffusion equation Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-12-06 Kin Ming Hui, Soojung Kim
Let $n \geq 3$, $0 \lt m \lt \frac{n-2}{n}$ and $T \gt 0$. We construct positive solutions to the fast diffusion equation $u_t = \Delta u^m$ in $\mathbb{R}^n \times (0, T)$, which vanish at time $T$. By introducing a scaling parameter $\beta$ inspired by $\href{https://dx.doi.org/10.4310/CAG.2019.v27.n8.a4}{\textrm{[DKS]}}$, we study the second-order asymptotics of the self-similar solutions associated
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On the existence of closed biconservative surfaces in space forms Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-12-06 S. Montaldo, A. Pámpano
Biconservative surfaces of Riemannian $3$-space forms $N^3(\rho)$, are either constant mean curvature (CMC) surfaces or rotational linear Weingarten surfaces verifying the relation $3 \kappa_1 + \kappa_2 = 0$ between their principal curvatures $\kappa_1$ and $\kappa_2$. We characterise the profile curves of the non-CMC biconservative surfaces as the critical curves for a suitable curvature energy.
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A Bourgain–Brezis–Mironescu–Dávila theorem in Carnot groups of step two Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-12-06 Nicola Garofalo, Giulio Tralli
In this note we prove the following theorem in any Carnot group of step two $\mathbb{G}$:\[\lim_{s \nearrow 1/2} (1 - 2s) \mathfrak{P}_{H,s} (E) = \frac{4}{\sqrt{\pi}} \mathfrak{P}_H (E).\]Here, $\mathfrak{P}_H (E)$ represents the horizontal perimeter of a measurable set $E \subset \mathbb{G}$, whereas the nonlocal horizontal perimeter $\mathfrak{P}_{H,s} (E)$ is a heat based Besov seminorm. This result
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Steklov eigenvalue problem on subgraphs of integer lattices Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-12-06 Wen Han, Bobo Hua
We study the eigenvalues of the Dirichlet-to-Neumann operator on a finite subgraph of the integer lattice $\mathbb{Z}^n$. We estimate the first $n + 1$ eigenvalues using the number of vertices of the subgraph. As a corollary, we prove that the first non-trivial eigenvalue of the Dirichlet-to-Neumann operator tends to zero as the number of vertices of the subgraph tends to infinity.
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Divide knots of maximal genus defect Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-12-06 Livio Liechti
We construct divide knots with arbitrary smooth four-genus but topological four-genus equal to one. In particular, for strongly quasipositive fibred knots, the ratio between the topological and the smooth four-genus can be arbitrarily close to zero.
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Small knots of large Heegaard genus Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-12-06 William Worden
Building off ideas developed by Agol, we construct a family of hyperbolic knots $K_n$ whose complements contain no closed incompressible surfaces and have Heegaard genus exactly $n$. These are the first known examples of such knots. Using work of Futer and Purcell, we are able to bound the crossing number for each $K_n$ in terms of $n$.
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Confined Willmore energy and the area functional Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-12-06 Marco Pozzetta
We consider minimization problems of functionals given by the difference between the Willmore functional of a closed surface and its area, when the latter is multiplied by a positive constant weight $\Lambda$ and when the surfaces are confined in the closure of a bounded open set $\Omega \subset \mathbb{R}^3$. We explicitly solve the minimization problem in the case $\Omega = B_1$. We give a description
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Bergman functions and the equivalence problem of singular domains Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-12-06 Bingyi Chen, Stephen S.-T. Yau
In this article, we use the Bergman function, which is introduced by the second author in $\href{ https://dx.doi.org/10.4310/MRL.2004.v11.n6.a8}{[\textrm{Ya}]}$, to study the equivalence problem of bounded complete Reinhardt domains in the singular variety $\widetilde{V} = \lbrace (u_1, u_2, u_3, u_4) \in \mathbb{C}^4 \vert u_1 u_4 = u_2 u_3 \rbrace$.
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Real Higgs pairs and non-abelian Hodge correspondence on a Klein surface Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-12-06 Indranil Biswas, Luis Ángel Calvo, Oscar García-Prada
We introduce real structures on $L$-twisted Higgs pairs over a compact connected Riemann surface $X$ equipped with an antiholomorphic involution, where $L$ is a holomorphic line bundle on $X$ with a real structure, and prove a Hitchin–Kobayashi correspondence for the $L$-twisted Higgs pairs. Real $G^\mathbb{R}$-Higgs bundles, where $G^\mathbb{R}$ is a real form of a connected semisimple complex affine
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Boundary unique continuation for the Laplace equation and the biharmonic operator Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-09-21 S. Berhanu
We establish results on unique continuation at the boundary for the solutions of $\Delta u = f$, $f$ harmonic, and the biharmonic equation $\Delta^2 u = 0$. The work is motivated by analogous results proved for harmonic functions by X. Huang et al in [$\href{https://doi.org/10.1080/03605309308820929}{\textrm{HK}}$] and [$\href{https://doi.org/10.1080/17476939508814787}{\textrm{HKMP}}$] and by M. S
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Avoidance for set-theoretic solutions of mean-curvature-type flows Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-09-21 Or Hershkovits, Brian White
We give a self-contained treatment of set-theoretic subsolutions to flow by mean curvature, or, more generally, to flow by mean curvature plus an ambient vector field. The ambient space can be any smooth Riemannian manifold. Most importantly, we show that if two such set-theoretic subsolutions are initially disjoint, then they remain disjoint provided one of the subsolutions is compact; previously
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Asymptotic convergence for modified scalar curvature flow Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-09-21 Ling Xiao
In this paper, we study the flow of closed, star-shaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^\alpha \sigma^{1/2}_{2}$, where $\sigma^{1/2}_{2}$ is the normalized square root of the scalar curvature, $\alpha \geq 2$, and $r$ is the distance from points on the hypersurface to the origin. We prove that the flow exists for all time and the star-shapedness is preserved. Moreover, after normalization
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Motion of level sets by inverse anisotropic mean curvature Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-09-21 Francesco Della Pietra, Nunzia Gavitone, Chao Xia
In this paper we consider the weak formulation of the inverse anisotropic mean curvature flow, in the spirit of Huisken–Ilmanen [$\href{ https://dx.doi.org/10.4310/jdg/1090349447}{15}$]. By using approximation method involving Finsler‑$p$-Laplacian, we prove the existence and uniqueness of weak solutions.
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The moduli space of $S^1$-type zero loci for $\mathbb{Z}/2$-harmonic spinors in dimension $3$ Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-09-21 Ryosuke Takahashi
Let $M$ be a compact oriented $3$-dimensional smooth manifold. In this paper, we construct a moduli space consisting of pairs $(\Sigma,\psi)$ where $\Sigma$ is a $C^1$-embedding simple closed curve in $M$, $\psi$ is a $\mathbb{Z}/2$-harmonic spinor vanishing only on $\Sigma$, and ${\lVert \psi \rVert}_{L^2_1} \neq 0$. We prove that when $\Sigma$ is $C^2$, a neighborhood of $(\Sigma,\psi)$ in the moduli
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Existence and multiplicity of solutions for a class of indefinite variational problems Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-08-17 Claudianor O. Alves, Minbo Yang
In this paper we study the existence and multiplicity of solutions for the following class of strongly indefinite problems\[(P)_k \qquad\begin{cases}-\Delta u + V(x)u=A(x/k)f(u) \; \textrm{in} \; \mathbb{R}^N, \\u ∈ H^1(\mathbb{R}^N),\end{cases}\]where $N \geq 1$, $k \in \mathbb{N}$ is a positive parameter, $f : \mathbb{R } \to \mathbb{R}$ is a continuous function with subcritical growth, and $V, A
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A generating function of a complex Lagrangian cone in $\mathbf{H}^n$ Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-08-17 Norio Ejiri
We formulate the space of multivalued branched minimal immersions of compact Riemann surfaces of genus $\gamma \geq 2$ into $\mathbf{R}^n$, and show that it is a complex analytic set. If an irreducible component of the complex analytic set admits a non-degenerate critical point, then we construct a complex Lagrangian cone in $\mathbf{H}^{n \gamma}$ derived from the complex period map, and obtain its
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Ancient solutions to the Ricci flow in higher dimensions Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-08-17 Xiaolong Li, Yongjia Zhang
In this paper, we study $\kappa$-noncollapsed ancient solutions to the Ricci flow with nonnegative curvature operator in higher dimensions $n \geq 4$. We impose one further assumption: one of the asymptotic shrinking gradient Ricci solitons is the standard cylinder $\mathbb{S}^{n-1} \times \mathbb{R}$. First, Perelman’s structure theorem on three-dimensional ancient $\kappa$-solutions is generalized
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On positive scalar curvature bordism Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-08-17 Paolo Piazza, Thomas Schick, Vito Felice Zenobi
Using standard results from higher (secondary) index theory, we prove that the positive scalar curvature bordism groups $\mathrm{Pos}^\operatorname{spin}_{4n} (G \times \mathbb{Z})$ are infinite for any $n \geq 1$ and $G$ a group with nontrivial torsion. We construct representatives of each of these classes which are connected and with fundamental group $G \times \mathbb{Z}$. We get the same result
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A heat flow problem from Ericksen’s model for nematic liquid crystals with variable degree of orientation, II Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-08-17 Chi-Cheung Poon
We study a heat flow problem for nematic liquid crystals with variable degree of orientation. Let $\Omega$ be a bounded domain in $\mathbb{R}^m$ with smooth boundary and $\mathcal{C}$ be the round cone in $\mathbb{R}^ \times \mathbb{R}^3$,\[\mathcal{C} = {\lbrace (s, u) \in \mathbb{R}^ \times \mathbb{R}^3 \; : \quad s^2 = {\lvert u \rvert}^2 \rbrace} \textrm{.}\]Under certain conditions on the double-well
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K3 surfaces with a pair of commuting non-symplectic involutions Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-08-17 Frank Reidegeld
We study K3 surfaces with a pair of commuting involutions that are non-symplectic with respect to two anti-commuting complex structures that are determined by a hyper-Kähler metric. One motivation for this paper is the role of such $\mathbb{Z}^2_2$-actions for the construction of Riemannian manifolds with holonomy $G_2$. We find a large class of smooth K3 surfaces with such pairs of involutions. After
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Asymptotic behaviour for the heat equation in hyperbolic space Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-08-17 Juan Luis Vázquez
Following the classical result of long-time asymptotic convergence towards a multiple of the Gaussian kernel that holds true for integrable solutions of the Heat Equation posed in the Euclidean Space $\mathbb{R}^n$, we examine the question of long-time behaviour of the Heat Equation in the Hyperbolic Space $\mathbb{H}^n, n \gt 1$, also for integrable data and solutions. We show that the typical convergence
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Guan–Li type mean curvature flow for free boundary hypersurfaces in a ball Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-08-17 Guofang Wang, Chao Xia
In this paper we introduce a Guan–Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan–Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.
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Singularities of axially symmetric volume preserving mean curvature flow Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-07-13 Maria Athanassenas, Sevvandi Kandanaarachchi
We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows—surfaces of revolution—in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additional curvature assumptions being imposed, while axial symmetry and boundary conditions
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Geometric wave propagator on Riemannian manifolds Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-07-13 Matteo Capoferri, Michael Levitin, Dmitri Vassiliev
We study the propagator of the wave equation on a closed Riemannian manifold $M$. We propose a geometric approach to the construction of the propagator as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. This enables us to provide a global invariant definition of the full symbol of the propagator—a scalar function on the cotangent bundle—and
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Plateau’s problem for singular curves Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-07-13 Paul Creutz
We give a solution of Plateau’s problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau’s problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this paper seems to be new even in $\mathbb{R}^n$.
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Stochastically complete submanifolds with parallel mean curvature vector field in a Riemannian space form Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-07-13 Henrique F. de Lima, Fábio R. dos Santos
In this paper, we deal with stochastically complete submanifolds $M^n$ immersed with nonzero parallel mean curvature vector field in a Riemannian space form $\mathbb{Q}^{n+p}_c$ of constant sectional curvature $c \in {\lbrace -1, 0, 1 \rbrace}$. In this setting, we use the weak Omori–Yau maximum principle jointly with a suitable Simons type formula in order to show that either such a submanifold $M^n$
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New surfaces with canonical map of high degree Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-07-13 Christian Gleissner, Roberto Pignatelli, Carlos Rito
We give an algorithm that, for a given value of the geometric genus $p_g$, computes all regular product-quotient surfaces with abelian group that have at most canonical singularities and have canonical system with at most isolated base points. We use it to show that there are exactly two families of such surfaces with canonical map of degree $32$. We also construct a surface with $q = 1$ and canonical
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Complex structures of a twenty-dimensional family of Calabi–Yau $3$-folds Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-07-13 Chuangqiang Hu, Stephen S.-T. Yau, Huaiqing Zuo
In this paper, we classify all isomorphic classes of a family of Calabi–Yau $3$-folds with $20$ parameters. In addition, we show that the isomorphisms form a finite group. The invariants under the action of this group are calculated by introducing the so-called DS‑graph.
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Li–Yau inequality on virtually Abelian groups Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-07-13 Gabor Lippner, Shuang Liu
We show that Cayley graphs of virtually Abelian groups satisfy a Li–Yau type gradient estimate despite the fact that they do not satisfy any known variant of the curvature-dimension inequality with non-negative curvature.
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Homogeneous metrics with prescribed Ricci curvature on spaces with non-maximal isotropy Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-07-13 Mark Gould, Artem Pulemotov
Consider a compact Lie group $G$ and a closed subgroup $H \lt G$. Suppose $\mathcal{M}$ is the set of $G$-invariant Riemannian metrics on the homogeneous space $M = G/H$. We obtain a sufficient condition for the existence of $g \in \mathcal{M}$ and $c \gt 0$ such that the Ricci curvature of $g$ equals $cT$ for a given $T \in \mathcal{M}$. This condition is also necessary if the isotropy representation
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Instability of some Riemannian manifolds with real Killing spinors Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-07-13 Changliang Wang, McKenzie Y. Wang
We prove the instability of some families of Riemannian manifolds with non-trivial real Killing spinors. These include the invariant Einstein metrics on the Aloff–Wallach spaces $N_{k,l} = \mathrm{SU}(3) / i_{k,l} (S^1)$ (which are all nearly parallel $\mathrm{G}_2$ except $N_{1,0}$), and Sasaki Einstein circle bundles over certain irreducible Hermitian symmetric spaces. We also prove the instability
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Positive mass theorem for initial data sets with corners along a hypersurface Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-05-25 Aghil Alaee, Shing-Tung Yau
We prove positive mass theorem with angular momentum and charges for axially symmetric, simply connected, maximal, complete initial data sets with two ends, one designated asymptotically flat and the other either (Kaluza–Klein) asymptotically flat or asymptotically cylindrical, for $4$-dimensional Einstein–Maxwell theory and $5$-dimensional minimal supergravity theory which metrics fail to be $C^1$
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Longtime existence of Kähler–Ricci flow and holomorphic sectional curvature Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-05-25 Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam, Freid Tong
In this work, we obtain some sufficient conditions for the longtime existence of the Kähler–Ricci flow solution. Using the existence results, we generalize a result by Wu–Yau on the existence of Kähler–Einstein metric on noncompact complex manifolds.
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An anisotropic shrinking flow and $L_p$ Minkowski problem Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-05-25 Weimin Sheng, Caihong Yi
In this paper, we consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in Euclidean $R^{n+1}$ with speed $f u^\alpha \sigma^{-\beta}_n$, where $u$ is the support function of the hypersurface, $\alpha , \beta \in R^1$, and $\beta \gt 0, \sigma_n$ is the $n$-th symmetric polynomial of the principle curvature radii of the hypersurface. We prove that the flow exists a unique smooth
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Positivity preserving along a flow over projective bundle Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-05-25 Xueyuan Wan
In this paper, we introduce a flow over the projective bundle $p : P(E^\ast) \to M$, a natural generalization of both Hermitian–Yang–Mills flow and Kähler–Ricci flow. We prove that the semi-positivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^\ast)} (1)$ is preserved along this flow under the null eigenvector assumption. As applications, we prove that the semi-positivity is preserved
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Generalized Kähler Taub-NUT metrics and two exceptional instantons Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-05-25 Brian Weber
We study the one-parameter family of generalized Kahler Taub-NUT metrics (discovered by Donaldson), along with two exceptional Taub-NUT-like instantons, and understand them to the extent that should be sufficient for blow-up and gluing arguments. In particular we parameterize their geodesics from the origin, determine curvature fall-off rates and volume growth rates for metric balls, and find blow-down
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Rectifiability and Minkowski bounds for the zero loci of $\mathbb{Z}/2$ harmonic spinors in dimension $4$ Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-05-25 Boyu Zhang
This article proves that the zero locus of a $\mathbb{Z}/2$ harmonic spinor on a $4$-dimensional manifold is $2$-rectifiable and has locally finite Minkowski content.
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$J$-holomorphic curves from closed $J$-anti-invariant forms Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-04-26 Louis Bonthrone, Weiyi Zhang
We study the relation between $J$-anti-invariant $2$-forms and pseudo-holomorphic curves in this paper. We show the zero set of a closed $J$-anti-invariant $2$-form on an almost complex $4$-manifold supports a $J$-holomorphic subvariety in the canonical class. This confirms a conjecture of Draghici–Li–Zhang. A higher dimensional analogue is established. We also show the dimension of closed $J$-anti-invariant
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Effective operators for changing sign Robin Laplacian in thin two- and three-dimensional curved waveguides Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-04-26 César R. de Oliveira, Alex F. Rossini
We study the Laplacian in some thin curved domains, in the plane and space, with particular types of Robin boundary conditions and cross-sections. We derive, when the diameters of the cross sections tend to zero, nontrivial effective Schrödinger operators on the reference curve by means of norm resolvent convergences. Besides the changing sign in the Robin parameter, for which no renormalization is
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Broken ray tensor tomography with one reflecting obstacle Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-04-26 Joonas Ilmavirta, Gabriel P. Paternain
We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and vanishes on the rest. This is done in a geometry with non-positive sectional curvature and a strictly convex obstacle in any dimension.We give two proofs, both of
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Convergence of energy functionals and stability of lower bounds of Ricci curvature via metric measure foliation Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-04-26 Daisuke Kazukawa
The notion of the metric measure foliation was introduced by Galaz-García, Kell, Mondino, and Sosa in [9]. They studied the relation between a metric measure space with a metric measure foliation and its quotient space. They showed that the curvature-dimension condition and the Cheeger energy functional preserve from a such space to its quotient space. Via the metric measure foliation, we investigate
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Extremally Ricci pinched $G_2$-structures on Lie groups Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-04-26 Jorge Lauret, Marina Nicolini
Only two examples of extremally Ricci pinched $G_2$-structures can be found in the literature and they are both homogeneous. We study in this paper the existence and structure of such very special closed $G_2$-structures on Lie groups. Strong structural conditions on the Lie algebra are proved to hold. As an application, we obtain three new examples of extremally Ricci pinched $G_2$-structures and
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Limiting case of an isoperimetric inequality with radial densities and applications Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-04-26 Georgios Psaradakis
We prove a sharp isoperimetric inequality with radial densities whose functional counterpart corresponds to a limiting case for the exponents of the Il’in (or Caffarelli–Kohn–Nirenberg) inequality in $L^1$. We show how the latter applies to obtain an optimal critical Sobolev weighted norm improvement to one of the $L^1$ weighted Hardy inequalities of [29]. Further applications include an $L^p$ version
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Degeneration of globally hyperbolic maximal anti-de Sitter structures along rays Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-04-26 Andrea Tamburelli
Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle of the Teichmüller space of a closed surface $S$, we study how some geometric quantities, such as the Lorentzian Hausdorff dimension of the limit set, the width of the convex core and the Hölder exponent, degenerate along rays of cotangent vectors.
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Macroscopic stability and simplicial norms of hypersurfaces Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-03-17 Hannah Alpert
We introduce a $Z$-coefficient version of Guth’s macroscopic stability inequality for almost-minimizing hypersurfaces. In manifolds with a lower bound on macroscopic scalar curvature, we use the inequality to prove a lower bound on areas of hypersurfaces in terms of the Gromov simplicial norm of their homology classes. We give examples to show that a very positive lower bound on macroscopic scalar
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On Gauduchon connections with Kähler-like curvature Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-03-17 Daniele Angella, Antonio Otal, Luis Ugarte, Raquel Villacampa
We study Hermitian metrics with a Gauduchon connection being “Kähler-like”, namely, satisfying the same symmetries for curvature as the Levi–Civita and Chern connections. In particular, we investigate $6$-dimensional solvmanifolds with invariant complex structures with trivial canonical bundle and with invariant Hermitian metrics. The results for this case give evidence for two conjectures that are
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Double branched covers of knotoids Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-03-17 Agnese Barbensi, Dorothy Buck, Heather A. Harrington, Marc Lackenby
By using double branched covers, we prove that there is a $1\textrm{-}1$ correspondence between the set of knotoids in $S^2$, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence allows us to study knotoids through tools and invariants coming from knot theory. In particular, concepts from geometrisation generalise to knotoids, allowing us
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Stability and area growth of $\lambda$-hypersurfaces Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-03-17 Qing-Ming Cheng, Guoxin Wei
In this paper, We define a $\mathcal{F}$-functional and study $\mathcal{F}$-stability of $\lambda$-hypersurfaces, which extend a result of Colding–Minicozzi [6]. Lower bound growth and upper bound growth of area for complete and non-compact $\lambda$-hypersurfaces are studied.
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A softer connectivity principle Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-03-17 Luis Guijarro, Frederick Wilhelm
We give soft, quantitatively optimal extensions of the classical Sphere Theorem, Wilking’s connectivity principle and Frankel’s Theorem to the context of Rick curvature. The hypotheses are soft in the sense that they are satisfied on sets of metrics that are open in the $C^2$-topology.
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The Calderón problem for the conformal Laplacian Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-03-17 Matti Lassas, Tony Liimatainen, Mikko Salo
We consider a conformally invariant version of the Calderón problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main result states that a locally conformally real-analytic manifold in dimensions $\geq 3$ can be determined in this way, giving a positive answer to an earlier conjecture
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Level curves of minimal graphs Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-03-17 Allen Weitsman
We consider minimal graphs $u = u(x, y) \gt 0$ over domains $D \subset R^2$ bounded by an unbounded Jordan arc $\gamma$ on which $u = 0$.We prove an inequality on the curvature of the level curves of $u$, and prove that if $D$ is concave, then the sets $u(x, y) \gt C (C \gt 0)$ are all concave. A consequence of this is that solutions, in the case where $D$ is concave, are also superharmonic.
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Weighted extremal Kähler metrics and the Einstein–Maxwell geometry of projective bundles Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-01-30 Vestislav Apostolov, Gideon Maschler, Christina W. Tønnesen-Friedman
We study the existence of weighted extremal Kähler metrics in the sense of [4, 32] on the total space of an admissible projective bundle over a Hodge Kähler manifold of constant scalar curvature. Admissible projective bundles have been defined in [5], and they include the projective line bundles [29] and their blow-downs [31], thus providing a most general setting for extending the existence theory
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Quasi-local mass on unit spheres at spatial infinity Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-01-30 Po-Ning Chen, Mu-Tao Wang, Ye-Kai Wang, Shing-Tung Yau
In this note, we compute the limit of the Wang–Yau quasi-local mass on unit spheres at spatial infinity of an asymptotically flat initial data set. Similar to the small sphere limit of the Wang–Yau quasi-local mass, we prove that the leading order term of the quasi-local mass recovers the stress-energy tensor. For a vacuum spacetime, the quasi-local mass decays faster and the leading order term is
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Eigenvalues upper bounds for the magnetic Schrödinger operator Commun. Anal. Geom. (IF 0.7) Pub Date : 2023-01-30 Bruno Colbois, Ahmad El Soufi, Saïd Ilias, Alessandro Savo
We study the eigenvalues $\lambda_k (H_{A,q})$ of the magnetic Schrödinger operator $ H_{A,q}$ associated with a magnetic potential $A$ and a scalar potential $q$, on a compact Riemannian manifold $M$, with Neumann boundary conditions if $\partial M \neq \emptyset$. We obtain various bounds on $\lambda_1 (H_{A,q}),\lambda_2 (H_{A,q})$ and, more generally, on $\lambda_k (H_{A,q})$. Some of them are