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Vertex scaling of piecewise linear metrics on surfaces introduced by Luo (Commun Contemp Math 6: 765–780, 2004) is a straightforward discretization of smooth conformal structures on surfaces. Combinatorial α -curvature for vertex scaling of piecewise linear metrics on surfaces is a discretization of Gaussian curvature on surfaces. In this paper, we investigate the prescribing combinatorial α -curvature problem on polyhedral surfaces. Using Gu-Luo-Sun-Wu's discrete conformal theory (J. Differ. Geom. 109: 223–256, 2018) for piecewise linear metrics on surfaces and variational principles with constraints, we prove some Kazdan-Warner type theorems for prescribing combinatorial α -curvature problem, which generalize the results obtained in Gu-Luo-Sun-Wu (J. Differ. Geom. 109: 223–256, 2018), Xu (Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I. arXiv:1806.04516v2) on prescribing combinatorial curvatures on surfaces. Gu-Luo-Sun-Wu (J. Differ. Geom. 109: 223–256, 2018) conjectured that one can prove Kazdan-Warner's theorems in Kazdan (Ann Math 99: 14–47, 1974), Kazdan (Ann Math 101: 317-331, 1975) via approximating smooth surfaces by polyhedral surfaces. This paper takes the first step in this direction. [ABSTRACT FROM AUTHOR]

In this paper, we study vacuum static spaces. We firstly derive a Bochner-type formula for the Weyl tensor to vacuum static space. Based on a global argument, under the condition of zero radial Weyl curvature, we then obtain a pointwise identity and use it to prove that each closed vacuum static space of dimension n ≥ 5 with scalar curvature n (n - 1) and zero radial Weyl curvature is Bach flat. [ABSTRACT FROM AUTHOR]

In Glickenstein (J Differ Geom 87: 201–237, 2011), Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. It includes Thurston's circle packings, Bowers–Stephenson's inversive distance circle packings and Luo's vertex scalings as special cases. In this paper, we study the deformation of Glickenstein's discrete conformal structures by combinatorial curvature flows. The combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces (Zhang et al. in Graph Models 76: 321–339, 2014) is a generalization of Chow–Luo's combinatorial Ricci flow for Thurston's circle packings (Chow and Luo in J Differ Geom 63: 97–129, 2003) and Luo's combinatorial Yamabe flow for vertex scalings (Luo in Commun Contemp Math 6: 765–780, 2004). We prove that the solution of the combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces can be uniquely extended. Furthermore, under some necessary conditions, we prove that the solution of the extended combinatorial Ricci flow on a triangulated surface exists for all time and converges exponentially fast for any initial value. We further introduce the combinatorial Calabi flow for Glickenstein's discrete conformal structures on triangulated surfaces and study the basic properties of the flow. These combinatorial curvature flows provide effective algorithms for finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures. [ABSTRACT FROM AUTHOR]

This paper is the continuation of the previous one Jiao and Cui (Area-Minimizing Cones Over Grassmannian Manifolds. J. Geom. Anal. 32, 224 (2022). https://doi.org/10.1007/s12220-022-00963-7), where we re-proved the area-minimization of cones over Grassmannians of n-planes G (n , m ; F) (F = R , C , H) , Cayley plane O P 2 from the point view of Hermitian orthogonal projectors, and gave area-minimizing cones associated to oriented real Grassmannians G ~ (n , m ; R) by using Lawlor's Curvature Criterion Lawlor (Mem Amer Math Soc 91(446), 1991). Here, we make a further step on showing that the cones, of dimension no less than 8 , over minimal products of G (n , m ; F) are area-minimizing. Moreover, those cones are very similar to the classical cones over products of spheres, and for the critical situation—the cones of dimension 7 Lawlor (Mem Amer Math Soc 91(446), 1991), we gain more area-minimizing cones by carefully computing the Jacobian i n f v d e t (I - t H ij v) . Certain minimizing cones among them had been found from the perspective of R-spaces Ohno and Sakai (Josai Math Monogr 13:69–91, 2021), or isoparametric theory Tang and Zhang (J Differ Geom 115(2):367–393, 2020) recently, and the generic ones in our results are completely new. We also prove that the cones over minimal product of general G ~ (n , m ; R) are area-minimizing, it can be seen as generalized results for some G ~ (2 , m ; R) shown in Ohno and Sakai (Josai Math Monogr 13:69–91, 2021), Tang and Zhang (J Differ Geom 115(2):367–393, 2020). [ABSTRACT FROM AUTHOR]

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in R n + 1 for all n ≥ 3 : we show that if a mean curvature flow { M t } in R n + 1 has an S n - 1 × R singularity at (x 0 , t 0) , then there exists an ε = ε (x 0 , t 0) > 0 such that M t ∩ B (x 0 , ε) is mean-convex for all t ∈ (t 0 - ε 2 , t 0 + ε 2) . As in the case n = 2 , which was resolved by the first three authors in Choi et al. (Acta Math, 2018), the existence of such a mean-convex neighborhood follows from classifying a certain class of ancient Brakke flows that arise as potential blowup limits near a neck singularity. Specifically, we prove that any ancient unit-regular integral Brakke flow with a cylindrical blowdown must be either a round shrinking cylinder, a translating bowl soliton, or an ancient oval. In particular, combined with a prior result of the last two authors (Hershkovits and White in Commun Pure Appl Math 73(3):558–580, 2020), we obtain uniqueness of mean curvature flow through neck singularities. The main difficulty in addressing the higher dimensional case is in promoting the spectral analysis on the cylinder to global geometric properties of the solution. Most crucially, due to the potential wide variety of self-shrinking flows with entropy lower than the cylinder when n ≥ 3 , smoothness does not follow from the spectral analysis by soft arguments. This precludes the use of the classical moving plane method to derive symmetry. To overcome this, we introduce a novel variant of the moving plane method, which we call "moving plane method without assuming smoothness"—where smoothness and symmetry are established in tandem. [ABSTRACT FROM AUTHOR]

This paper is the first of two parts which considers the rolling (or development) of two Riemannian connected manifolds (M,g) and M ̂ , ĝ of dimensions 2 and 3 respectively, with the constraints of no-spinning and no-slipping. The present work is a continuation of Mortada et al. (Acta Appl Math 139:105–31, 2015), which modeled the general setting of the rolling of two Riemannian connected manifolds with different dimensions as a driftless control affine system on a fibered space Q of dimension eighth, with an emphasis on understanding the local structure of the rolling orbits, i.e., the reachable sets in Q. We show that the possible dimensions of non open rolling orbits belong to the set {2, 5, 6, 7}. In this first part, we describe the structures of orbits of dimension 2, one of the two possible local structure of rolling orbits of dimension 5 and special cases of dimension 7. [ABSTRACT FROM AUTHOR]

We develop the asymptotic analysis as ε→0 for the natural gradient flow of the self-dual U(1)-Higgs energies on Hermitian line bundles over closed manifolds (Mn,g) of dimension n≥3, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows—i.e., integral (n−2)-Brakke flows—generalizing results of (Pigati and Stern in Invent. Math. 223:1027–1095, 2021) from the stationary case. Given any integral (n−2)-cycle Γ0 in M, these results can be used together with the convergence theory developed in (Parise et al. in Convergence of the self-dual U(1)-Yang–Mills–Higgs energies to the (n−2)-area functional, 2021, arXiv:2103.14615) to produce nontrivial integral Brakke flows starting at Γ0 with additional structure, similar to those produced via Ilmanen's elliptic regularization. [ABSTRACT FROM AUTHOR]

We consider overdetermined problems for Hessian quotient equations and Hessian quotient curvature equations, which are fully nonlinear elliptic equations. We establish Rellich–Pohozaev-type identities for Hessian quotient equations and Hessian quotient curvature equations. Based on these identities and the maximum principle for P functions, the symmetry of solutions can be proved in the Euclidean space. We also prove the related result for Hessian quotient equations in the hyperbolic space. Our results generalize the overdetermined problems for k-Hessian equations and k-curvature equations. [ABSTRACT FROM AUTHOR]

We consider the following fractional prescribed scalar curvature problem where N > 2 s + 2 , 1 2 < s < 1 , 2 s ∗ = 2 N N - 2 s is the critical Sobolev exponent, H ˙ s (R N) is the completion of C 0 ∞ (R N) under the semi-norm [ u ] H s (R N) : = (∫ R N | (- Δ) s 2 u | 2 d x) 1 2 , and K(y) is a positive radial function. We first prove the non-degeneracy result for the positive bubble solutions of the above equation via the local Pohozaev identities. Then we apply the non-degeneracy result to construct new bubble solutions by finite-dimensional reduction method. It should be mentioned that due to the non-localness of the fractional operator, we should establish the local Pohozaev identities for corresponding harmonic extension instead of u. This difference not only makes the sharp estimates for several integrals in the local Pohozaev identities, but also forces us to use the Pohozaev identities in quite a different way. [ABSTRACT FROM AUTHOR]

We establish a pointwise estimate of | A | along the mean curvature flow in terms of the initial geometry and the | H A | bound. As corollaries we obtain the blowup rate estimate of | H A | and an extension theorem with respect to | H A | . [ABSTRACT FROM AUTHOR]

We prove the existence of Sobolev extension operators for certain uniform classes of domains in a Riemannian manifold with an explicit uniform bound on the norm depending only on the geometry near their boundaries. We use this quantitative estimate to obtain uniform Neumann heat kernel upper bounds and gradient estimates for positive solutions of the Neumann heat equation assuming integral Ricci curvature conditions and geometric conditions on the possibly non-convex boundary. Those estimates also imply quantitative lower bounds on the first Neumann eigenvalue of the considered domains. [ABSTRACT FROM AUTHOR]

Motivated by Luo's (Commun Contemp Math 6:765–780, 2004) combinatorial Yamabe flow on closed surfaces and Guo's (Commun Contemp Math 13:827–842, 2011) combinatorial Yamabe flow on surfaces with boundary, we introduce combinatorial Calabi flow on ideally triangulated surfaces with boundary, aiming at finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths. Then we prove the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary. We further introduce fractional combinatorial Calabi flow on surfaces with boundary, which unifies and generalizes the combinatorial Yamabe flow and the combinatorial Calabi flow on surfaces with boundary. The long time existence and global convergence of fractional combinatorial Calabi flow are also proved. These combinatorial curvature flows provide effective algorithms to construct hyperbolic surfaces with totally geodesic boundaries with prescribed lengths. [ABSTRACT FROM AUTHOR]

We prove some reverse Laplacian comparison and relative volume comparison results under the situation where one has an integral bound for the part of the Ricci curvature which lies above a prescribed continuous function of the distance parameter. These extend parts of results of Ding (Chin Ann Math Ser B 15(1):35–42, 1994) and Kura (Proc Jpn Acad Ser A Math Sci 78(1):7–9, 2002) from pointwise Ricci curvature to integral Ricci curvature. [ABSTRACT FROM AUTHOR]

Based on the asymptotic property of the level hypersurfaces of f, we show that the solvability of Dirichlet problem for the fully nonlinear elliptic equation with Γ = Γ n is closely related to the existence of a C -subsolution introduced by Székelyhidi (J Differ Geom 109: 337–378, 2018) of a rescaled equation. For the complex Monge–Ampère equation and complex Hessian equations the gradient estimate established in previous works (Błocki, Math Ann 344: 317–327, 2009; Hanani, J Funct Anal 137: 49–75, 1996; Guan and Li, Adv Math 225: 1185–1223, 2010; Zhang, Int Math Res Not 2010: 3814–3836, 2010) also follows as a consequence of our argument. Also, the existence and regularity of admissible solutions to Dirichlet problem for fully nonlinear elliptic equations on compact Kähler manifolds of nonnegative orthogonal bisectional curvature are obtained. [ABSTRACT FROM AUTHOR]