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This article concerns the L p {L}_{p} Minkowski problem for q-capacity. We consider the case p ≥ 1 p\ge 1 and 1 < q < n 1\lt q\lt n in the smooth category by a kind of curvature flow, which converges smoothly to the solution of a Monge-Ampére type equation. We show the existence of smooth solution to the problem for p ≥ n p\ge n . We also provide a proof for the weak solution to the problem when p ≥ 1 p\ge 1 , which has been obtained by Zou and Xiong.

We derive interior curvature bounds for admissible solutions of a class of mixed Hessian curvature equations subject to affine Dirichlet data. As an application, we study a Plateau type problem for locally convex Weingarten hypersurfaces.

In this paper we study the Lpq-dual Minkowski problem for the case p < 0 < q. We prove for any positive smooth function f on $$\mathbb{S}^{1}$$ , there exists an F: ℝ+ → ℝ−, such that if F(q) < p < 0 or 0 < q < −F(−p) then there is a smooth and strictly convex body solving the planar Lpq-dual Minkowski problem.

We consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_k^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_k is the k-th symmetric polynomial of the principle curvature radii of the hypersurface, k is an integer and 1= k+1 where p=2-alpha, which is identify with Ivaki's recent result., 23 pages. arXiv admin note: text overlap with arXiv:1905.04679

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^{\alpha} K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. If $\alpha \ge n+1$, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if $f \equiv 1$. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang (Acta Math. 216 (2016): 325-388), for the case $q0$, we also establish the same results for even function $f$ and origin-symmetric initial condition, but for non-symmetric $f$, counterexample is given for the above smooth convergence., Comment: 34 pages

In this paper, we study a class of contracting flows of closed, convex hypersurfaces in the Euclidean space $${\mathbb {R}}^{n+1}$$ with speed $$r^{\alpha } \sigma _k$$, where $$\sigma _k$$ is the k-th elementary symmetric polynomial of the principal curvatures, $$\alpha \in {\mathbb {R}}^1$$, and r is the distance from the hypersurface to the origin. If $$\alpha \ge k+1$$, we prove that the flow exists for all time, preserves the convexity and converges smoothly after normalisation to a sphere centred at the origin. If $$\alpha

To study the prescribed $k$-curvature problem of 2nd-order symmetric curvature tensors on complete noncompact Riemannian manifolds, we consider a class of fully nonlinear elliptic partial differential equations. It is proved that on a Riemannian manifold with negative sectional curvature and Ricci curvature bounded from below, the equation is solvable provided that all the eigenvalues of the tensor are negative. The result is applicable to the prescribed $k$-curvature problems of modified Schouten tensor and Bakry–Émery Ricci tensor.

In this paper, we study the rigidity problem on closed locally conformally flat manifolds with constant k-curvature or constant quotient curvature. We prove that such manifolds must have constant sectional curvature under the assumption that the manifolds have positive sectional curvature. The same result is also held on closed locally conformally flat manifolds with totally geodesic boundary.

In this paper, we study the critical metrics with cyclic parallel Ricci tensor for volume functional on manifolds with boundary. We show such metric is isometric to a geodesic ball in a simply connected space form $$\mathbb {R}^n$$ , $$\mathbb {H}^n$$ or $$\mathbb {S}^n$$ under some extra conditions.

In this paper, we investigate a class of quadratic Riemannian curvature functionals on closed smooth manifold M of dimension n ⩾ 3 on the space of Riemannian metrics on M consisting of metrics with unit volume. We study the stability of these functionals at the metric with constant sectional curvature as its critical point.

In this paper, we study the critical metrics for quadratic curvature functionals involving the Ricci curvature and scalar curvature in the space of Riemannian metrics with unit volume. For these functionals, Einstein metrics are always critical metrics. However, a converse problem is not always true. The purpose of this paper is to show that, under the condition that the critical metrics are Bach-flat, a partial converse is true.

In this paper, we consider the solitons to the fully nonlinear type Yamabe flows. We prove that any k -Yamabe soliton with positive k -curvature must have constant σ k -curvature on closed locally conformally flat manifolds. We also prove that any quotient Yamabe soliton with positive k and l curvatures must have constant σ k σ l -curvature on closed locally conformally flat manifolds. Similar results are also proved on compact locally conformally flat manifolds with totally geodesic boundary.

We consider the CR Yamabe flow on a compact strictly pseudoconvex CR manifold $M$ of real dimension $2n+1$. We prove convergence of the CR Yamabe flow when $n=1$ or $M$ is spherical., Accepted by Mathematische Annalen

The authors consider the problem of conformally deforming a metric such that the k-curvature defined by an elementary symmetric function of the eigenvalues of the Bakry-Emery Ricci tensor on a compact manifold with boundary to a prescribed function. A consequence of our main result is that there exists a complete metric such that the Monge-Ampere type equation with respect to its Bakry-Emery Ricci tensor is solvable, provided that the initial Bakry-Emery Ricci tensor belongs to a negative convex cone.

We consider the problem of conformally deforming a metric by a fully nonlinear flow on a compact manifold with totally geodesic boundary. We prove that the k -curvature defined by the modified Schouten tensor always can be deformed in a natural way to a constant at exponential rate and the boundary preserves totally geodesic.

In this paper we consider the problem of finding closed hypersurfaces with prescribed Weingarten curvature in a large class of Riemannian manifolds. Under some sufficient conditions, we obtain an existence result by establishing a priori estimates and standard degree theory for the admissible solutions to the prescribed curvature equation. We mainly show that the primary technique developed in Urbas, Wang and the second author’s work in 2004 is flexible enough to be used in more general ambient geometry.

In this paper we consider Yamabe type problem for higher order curvatures on manifolds with totally geodesic boundaries. We prove local gradient and second derivative estimates for solutions to the fully nonlinear elliptic equations associated with the problems.

1.1. The k-Yamabe problem. The k-Yamabe problem is a higher order extension of the celebrated Yamabe problem for scalar curvature. It was initially proposed by Viaclovsky [72] and also arose in the study of Q-curvatures in [11]. Viaclovsky found that in a conformal metric, the resultant k-curvature equation can be expressed as an equation similar to the k-Hessian equation, which has been studied by many authors [77]. The k-Yamabe problem can be formulated as follows. Let (Mn, g) be a closed Riemannian manifold of dimension n ≥ 3. It is well-known that there is an orthonormal decomposition of the Riemannian curvature tensor

In this paper, we consider the total mass of asymptotically flat Riemannian manifold under the connection Ricci flow, which can also be viewed as renormalization group flow for nonlinear sigma model. Using the maximum principle, we show that the asymptotically Euclidean property is preserved under this flow. Moreover, if the decay rate of the AE manifold is fast enough, then the ADM mass is invariant under the flow.

We study the problem of finding complete conformal metrics determined by some symmetric function of the modified Schouten tensor on compact manifolds with boundary; which reduces to a Dirichlet problem. We prove the existence of the solution under some suitable conditions. In particular, we prove that every smooth compact n-dimensional manifold with boundary, with n ≥ 3, admits a complete Riemannian metric g whose Ricci curvature Ricg and scalar curvature R g satisfy This result generalizes Aviles and McOwen's in the scalar curvature case.

In this paper we study the problem of conformally deforming a metric to a prescribed k-th order symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with totally geodesic boundary. We prove the solvability and the compactness of the solution set for the cases k ≥ n/2, provided the conformal class admits k-admissible metric. These results had been proved by Gursky and Viaclovsky, Trudinger and Wang for the manifolds without boundary, and by Jin, Li and Li, and S. Chen for the locally conformally flat manifolds with boundary.

This paper mainly considers the translating soliton of Hk-flow for k > 0. We give the asymptotic expression of the entire rotationally symmetric translating soliton, and obtain non-convex “Wing-like” solution as well as two barrier solutions. Moreover, we show that the solution with polynomial growth keeps its growth rate when evolution.

Let M n be a smooth, compact manifold without boundary, and F 0: M n → R n+1 a smooth immersion which is convex. The one-parameter families F(·, t): M n × [0, T) → R n+1 of hypersurfaces M = F(·, t)(M n ) satisfy an initial value problem dt/dF(·, t) = −H k (·, t)ν(·, t), F(·, 0) = F 0(·), where H is the mean curvature and ν(·, t) is the outer unit normal at F(·, t), such that −H ν = $$ \overrightarrow H $$ is the mean curvature vector, and k > 0 is a constant. This problem is called H k -flow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type I and Type II. It is proved that for Type I singularity, the limiting hypersurface satisfies an elliptic equation; for Type II singularity, the limiting hypersurface must be a translating soliton.

In most previous works on the existence of solutions to the σ k -Yamabe problem, one assumes that the initial metric go is k-admissible. This is a pointwise condition. In this paper we prove that this condition can be replaced by a weaker integral condition.

In this paper we study the long time existence of solutions for a class of fully nonlinear parabolic equations arising from conformal geometry. In particular we prove that every smooth compact n dimensional manifold, \(n\ge3\), admits a Riemannian metric g with its Ricci curvature Ric and scalar curvature R satisfying $$det(Ric-R\cdot g)=const..$$

In this paper we study the regularity of closed, convex surfaces which achieve maximal affine area among all the closed, convex surfaces enclosed in a given domain in the Euclidean 3-space. We prove the C 1, α regularity for general domains and C 1,1 regularity if the domain is uniformly convex.

By studying a fully nonlinear flow deforming conformal metrics on compact and connected manifold, we prove that for $\lambda < 1$ , any metric g with its modified Schouten tensor $A^\lambda_{g}\in \Gamma_k^-$ always can be deformed in a natural way to a conformal metric with constant $\sigma_k$ -scalar curvature at exponential rate.

In this paper we study the existence of closed convex hypersurfaces in the Euclidean space Rn+1 such that a Weingarten curvature, regarded as a function of their unit normal, is equal to a given positive function on the unit sphere Sn. Two classical problems of this type, namely the Christoffel problem [6] and the Minkowski problem [16], which address respectively the cases of harmonic and Gauss curvature, were completely solved in the 1970’s [7,9,17]. Besides the harmonic and Gauss curvatures, the most interesting Weingarten curvatures are probably the mean curvature, or more generally the k-curvatures (1 ≤ k ≤ n)

In this paper we prove that the affine diameter of any closed uniformly convex hypersurface in Euclidean space enclosing finite volume is bounded from above.

Some geometric behaviours of complete solutions to mean curvature flow before the singularities occur are studied. The author obtains the estimates of the rate of the distance between two fixed points and the derivatives of the second fundamental form. By use of a new maximum principle, some geometric properties at infinity are obtained.

We study the long-time behavior of the solution to the connection Ricci flow on closed three-manifolds. The connection Ricci flow is a generalization of the Ricci flow to connection with torsion. We compare the evolving metrics under the connection Ricci flow and Ricci flow for some special cases. We obtain the blow-up rate of the Ricci tensor under the connection Ricci flow. Based on the result, we give the classification of the maximal solutions to the flow.

We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial background manifold is an approximation of a spherical space form and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point in finite time or converge to a totally geodesic sphere as the time tends to infinity.

In this paper a flow of convex hypersurfaces in the Euclidean space by the linear combination of the mean curvature and the n-th root of the Gauss-Kronecker curvature is considered. It is proved that such deforming convex hypersurfaces converge to a round sphere in the Huisken's sense.

In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space $Bbb R^n+1$ with positive mean curvature is $kappa$-noncollapsing, and a blow-up sequence converges locally smoothly along a subsequence to a smooth, convex blow-up solution. As a consequence we obtain a local Harnack inequality for the mean convex flow.

Let M be a compact Riemannian manifold of dimension n > 2. The k-curvature, for k = 1,2,...,n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a conformal metric whose k-curvature is a constant. When k = 1, it reduces to the well-known Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions is known for the case k = 2, n = 4, for locally conformally flat manifolds and for the cases k > n/2. In this paper we prove the solvability of the k-Yamabe problem in the remaining cases k ≤ n/2, under the hypothesis that the problem is variational. This includes all of the cases k = 2 as well as the locally conformally flat case.

We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Monge-Ampère type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.

In this paper we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is non-negative definite.