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In this paper, by using Schwarz rearrangement and isoperimetric inequalities, we prove comparison results for the solutions of Poisson equations on complete Riemannian manifolds with $Ric\geq (n-1)\kappa$, $\, \kappa\geq 0$, which extends the results in \cite{ANT-Talenti-a}. Furthermore, as applications of our comparison results, we obtain the Saint-Venant inequality and Bossel-Daners inequality for Robin Laplacian., Comment: Comments are welcome

In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ ( θ ) ϕ ⁢ ( ρ ) α − 2 ⁢ ϕ ⁢ ( ρ ) 2 + | ∇ ¯ ⁢ ρ | 2 , K=\frac{\tilde{f}(\theta)}{\phi(\rho)^{\alpha-2}\sqrt{\phi(\rho)^{2}+\lvert\overline{\nabla}\rho\rvert^{2}}}, a generalization of the Alexandrov problem ( α = n + 1 \alpha=n+1 ) in hyperbolic space, where f ~ \tilde{f} is a smooth positive function on S n \mathbb{S}^{n} , 𝜌 is the radial function of the hypersurface, ϕ ⁢ ( ρ ) = sinh ⁡ ρ \phi(\rho)=\sinh\rho and 𝐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 \alpha\geq n+1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 \mathbb{H}^{n+1} . We also consider the cases 2 < α ≤ n + 1 2 under the evenness assumption of f ~ \tilde{f} and prove the existence of solutions to the above equations.

In this paper, we investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in $$\mathbb {R}^{n+1}$$ and $$\mathbb {S}^{n+1}$$ by $$\sigma _k^\alpha $$ , where $$\sigma _k$$ is the k-th elementary symmetric function of the principal curvatures and $$\alpha \ge 1/k$$ . We prove that for any $$n\ge 3$$ and any fixed k with $$1\le k\le n$$ , there exists a constant $$c(n,k)>1/k$$ such that if $$\alpha $$ lies in the interval [1/k, c(n, k)], then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal curvature at every point of the flow hypersurface, and we prove that the properly rescaled hypersurfaces converge exponentially to the unit sphere. In the case $$1

In this paper, we prove a family of identities for closed and strictly convex hypersurfaces in the sphere and hyperbolic/de Sitter space. As applications, we prove Blaschke-Santaló type inequalities in the sphere and hyperbolic/de Sitter space, which generalizes the previous work of Gao, Hug and Schneider \cite{GHS03}. We also prove the quermassintegral inequalities in hyperbolic/de Sitter space., 23 pages, no figures. All comments are appreciated

We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures. In particular, we define local angle functions encoding the geometry of the Lagrangian submanifold at hand. We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface. We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions, respectively all but one, coincide.

In this paper, we prove a family of sharp geometric inequalities for free boundary hypersurfaces in a ball in space forms., All comments are welcome

In this paper, we consider nonhomogeneous inverse Gauss curvature flows in hyperbolic space H 3 \mathbb {H}^{3} . We prove that if the initial hypersurface Σ 0 \Sigma _0 has nonnegative scalar curvature, then the evolving surface Σ t \Sigma _{t} has nonnegative scalar curvature along the flow for t > 0 t>0 . The solution of the flow exists for all time and becomes more and more umbilic as t → ∞ t\rightarrow \infty .

In this article, we prove Talenti's comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, $L^1$- and $L^\infty$-moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse H\"older inequality for eigenfunctions of Dirichlet Laplacian., Comment: 16 pages. Comments are welcome

In this paper, we investigate closed strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree $1$ and inverse concave function of the principal curvatures with power greater than $1$, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere $\mathbb{S}^{n+1}_{+}$ with power greater than or equal to $1$., 23 pages, revised version, some typos are corrected, to appear in Crelle's Journal

In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar curvature, then along the flow by a positive power $\alpha$ of the mean curvature $H$, the evolving surface $M_t$ has positive scalar curvature for $t>0$. By assuming $\alpha\in [1,4]$, we can further prove that $M_t$ contracts a point in finite time and become spherical as the final time is approached. We also show the same conclusion for the flows by powers of scalar curvature and by powers of Gauss curvature provided that the power $\alpha\in [1/2,1]$. In the sphere case, we show that the flow by a positive power $\alpha$ of mean curvature contracts strictly convex surface in $\mathbb{S}^3$ to a round point in finite time if $\alpha\in [1,5]$. The same conclusion also holds for the flow by powers of Gauss curvature provided that the power $\alpha\in [1/2,1]$., Comment: 29 pages, v3, accepted version for Calculus of Variations and Partial Differential Equations; v2, add more details in section 7

In this paper, we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li [7]. This flow preserves the $m$th quermassintegral and decreases $(m+1)$th quermassintegral, so the convergence of the flow yields sharp Alexandrov-Fenchel type inequalities in hyperbolic space. Some special cases have been studied in [7]. In the first part of this paper, we show that h-convexity of the hypersurface is preserved along the flow and then the smooth convergence of the flow for h-convex hypersurfaces follows. We then apply this result to establish some new sharp geometric inequalities comparing the integral of $k$th Gauss-Bonnet curvature of a smooth h-convex hypersurface to its $m$th quermassintegral (for $0\leq m\leq 2k+1\leq n$), and comparing the weighted integral of $k$th mean curvature to its $m$th quermassintegral (for $0\leq m\leq k\leq n$). In particular, we give an affirmative answer to a conjecture proposed by Ge, Wang and Wu in 2015. In the second part of this paper, we introduce a new locally constrained curvature flow using the shifted principal curvatures. This is natural in the context of h-convexity. We prove the smooth convergence to a geodesic sphere of the flow for h-convex hypersurfaces, and provide a new proof of the geometric inequalities proved by Andrews, Chen and the third author of this paper in 2018. We also prove a family of new sharp inequalities involving the weighted integral of $k$th shifted mean curvature for h-convex hypersurfaces, which as application implies a higher order analogue of Brendle, Hung and Wang's [8] inequality., 38 pages, accepted version for Mathematische Annalen, add Corollary 1.10 to describe the application of the new locally constrained flow (1.11)

Let ( M n , g ) (M^n,g) be an n n -dimensional ( n ≥ 4 ) (n\geq 4) compact locally conformally flat Riemannian manifold with constant scalar curvature and constant squared norm of Ricci curvature. Applying the moving frame method, we prove that such a Riemannian manifold does not exist if its Ricci curvature tensor has three distinct eigenvalues.

We study the stability of unduloids with free boundary in the domain B between two parallel hyperplanes in Rn+1. If the unduloid has one half of period in B and is sufficiently close to a cylinder, then for 2 ≤ n ≤ 10, it is unstable; while for n ≥ 11, it is stable. If the unduloid has two or more halves of period in B and is sufficiently close to a cylinder, then for all n ≥ 2, it is unstable.

In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow to prove a new class of geometric inequalities for horospherically convex hypersurfaces in hyperbolic space. Using these new Alexandrov-Fenchel type inequalities and the inverse mean curvature flow, we obtain an Alexandrov-Fenchel inequality for strictly convex hypersurfaces in hyperbolic space, which was previously proved for horospherically convex hypersurfaces by Wang and Xia [44]. Finally, we use the mean curvature flow to prove a new Heintze-Karcher type inequality for hypersurfaces with positive Ricci curvature in hyperbolic space., Comment: All comments are welcome!

In this paper, through making careful analysis of Gauss and Codazzi equations, we prove that four dimensional biharmonic hypersurfaces in nonzero space forms have constant mean curvature. Our result gives the positive answer to the conjecture proposed by Balmus–Montaldo–Oniciuc in 2008 for four dimensional hypersurfaces.

We prove that, in a Euclidean domain bounded by hyperplanes having linearly independent normals, a connected oriented compact immersed stable capillary hypersurface $$M^n$$ disjoint from the edges of the domain and with the contact angles belonging to $$[\pi /2,\pi ]$$ must be part of a sphere, if $$\partial M$$ is embedded for $$n=2$$ , or $$\partial M$$ is convex in the hyperplane for $$n\ge 3$$ . By applying a similar argument, we also discuss two other cases where a stable capillary hypersurface must be part of a sphere.

By adapting the test functions introduced by Choi–Daskaspoulos (Uniqueness of closed self-similar solutions to the Gauss curvature flow. arXiv:1609.05487 , 2016) and Brendle–Choi–Daskaspoulos (Acta Math 219(1):1–16, 2017) and exploring the properties of the k-th elementary symmetric function $$\sigma _{k}$$ intensively, we show that for any fixed k with $$1\le k\le n-1$$ , any strictly convex closed hypersurface in $${\mathbb {R}}^{n+1}$$ satisfying $$\sigma _{k}^{\alpha }=\langle X,\nu \rangle $$ , with $$\alpha \ge \frac{1}{k}$$ , must be a round sphere. In fact, we prove a uniqueness result for any strictly convex closed hypersurface in $${\mathbb {R}}^{n+1}$$ satisfying $$F+C=\langle X,\nu \rangle $$ , where F is a positive homogeneous smooth symmetric function of the principal curvatures and C is a constant.

By employing a nice adapted frame we prove a Bonnet-type existence and uniqueness theorem for almost complex surfaces in the homogeneous nearly Kahler manifold $$S^3\times S^3$$ . The proof uses a local correspondence between almost complex surfaces in $$S^3\times S^3$$ and surfaces in $$\mathbb {R}^3$$ that satisfy the Wente H-surface equation. Furthermore we give a complete classification of flat almost complex surfaces in the homogeneous nearly Kahler $$S^3\times S^3$$ .

Given a compact Riemannian manifold M, we consider a warped product manifold $${\bar{M}} = I \times _h M$$ , where I is an open interval in $${\mathbb {R}}$$ . For a positive function $$\psi $$ defined on $${\bar{M}}$$ , we generalize the arguments in Guan et al. (Commun. Pure Appl. Math. 68(8):1287–1325, 2015) and Ren and Wang (On the curvature estimates for Hessian equations, 2016. arXiv:1602.06535 ), to obtain the curvature estimates for Hessian equations $$\sigma _k(\kappa )=\psi (V,\nu (V))$$ . We also obtain some existence results for the starshaped compact hypersurface $$\Sigma $$ satisfying the above equation with various assumptions.

In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces $\Sigma$ with nonnegative sectional curvature in $\mathbb{H}^n$. As an application, we prove the hyperbolic Alexandrov-Fenchel inequalities for hypersurfaces with nonnegative sectional curvature in $\mathbb{H}^n$: \begin{align*} \int_{\Sigma} p_{2k}\geq \omega_{n-1}\left[\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^\frac{1}{k}+\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{1}{k}\frac{n-1-2k}{n-1}}\right]^k, \end{align*} where $p_i$ is the normalized $i$-th mean curvature. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\mathbb{H}^n$. For a domain $\Omega\subset \mathbb{H}^n$ with $\Sigma=\partial \Omega$ having nonnegative sectional curvature, we prove an optimal inequality for quermassintegral in $\mathbb{H}^n$: \begin{align*} W_{2k+1}(\Omega)\geq \frac{\omega_{n-1}}{n}\sum_{i=0}^{k}\frac{n-1-2k}{n-1-2i}C_k^i\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^\frac{n-1-2i}{n-1}, \end{align*} where $W_i(\Omega)$ is the $i$-th quermassintegral in integral geometry. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\mathbb{H}^n$. All these inequalities was previously proved by Ge, Wang and Wu \cite{Ge-Wang-Wu2014} under the stronger condition that $\Sigma$ is horospherical convex., Comment: 20 pages, all comments are welcome!

© 2015 Elsevier B.V. We give a complete classification of Lagrangian immersions of homogeneous 3-manifolds (the Berger spheres, the Heisenberg group Nil3, the universal covering of the Lie group PSL(2,R) and the Lie group Sol3) in 3-dimensional complex space forms. As a corollary, we get a new characterization of the Berger sphere in complex projective space. publisher: Elsevier articletitle: A new characterization of the Berger sphere in complex projective space journaltitle: Journal of Geometry and Physics articlelink: http://dx.doi.org/10.1016/j.geomphys.2015.02.009 content_type: article copyright: Copyright © 2015 Elsevier B.V. All rights reserved. ispartof: Journal of Geometry and Physics vol:92 pages:129-139 status: published

This paper is concerned with the fully nonlinear equation σ 2 ( g ) = a σ 1 ( g ) + b . The first result is to obtain the entire solutions of the equation for conformally flat metric on R n under some additional assumptions, which generalizes the famous result of Chang–Gursky–Yang in [3] . The second one is to classify compact locally conformally flat manifolds with nonnegative Ricci curvature and the metric g satisfying the equation.

© 2016 Elsevier B.V. It is well-known that Vrancken–Li–Simon classified locally strongly convex affine hyperspheres in Rn+1 whose affine metric are of constant sectional curvatures, but on the other side it is still a difficult problem to classify n-dimensional locally strongly convex affine hyperspheres whose affine metrics are Einstein. In this paper, we have solved the problem in case n=4. publisher: Elsevier articletitle: On four-dimensional Einstein affine hyperspheres journaltitle: Differential Geometry and its Applications articlelink: http://dx.doi.org/10.1016/j.difgeo.2016.10.003 content_type: article copyright: © 2016 Elsevier B.V. All rights reserved. ispartof: Differential Geometry and its Applications vol:50 pages:20-33 status: published

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere., Comment: 11 pages. All comments are welcome

We consider a conformally invariant functional of the space of Riemannian metrics, compute its first variational formula and obtain some examples of its critical point. Then we compute the second variation of the conformally invariant functional about the above examples and get some stability results. These results extend theorems of Kobayashi (J Math Soc Jpn 37:373–389, 1985).

We consider gradient estimates to positive solutions of porous medium equations and fast diffusion equations: $$u_t=\Delta_\phi(u^p)$$ associated with the Witten Laplacian on Riemannian manifolds. Under the assumption that the $m$-dimensional Bakry-Emery Ricci curvature is bounded from below, we obtain gradient estimates which generalize the results in [20] and [13]. Moreover, inspired by X. -D. Li's work in [19] we also study the entropy formulae introduced in [20] for porous medium equations and fast diffusion equations associated with the Witten Laplacian. We prove monotonicity theorems for such entropy formulae on compact Riemannian manifolds with non-negative $m$-dimensional Bakry-Emery Ricci curvature, Comment: 25 pages

We obtain a Calabi-Yau type lower volume growth estimates for complete noncompact self-shrinkers of the mean curvature flow, more precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth., 14 pages, we change the title of the paper, delete a theorem about Ricci soliton which has been proved by Ovidiu Munteanu and Jiaping Wang, and correct a gap in the proof of our theorem in the previous version

The totally umbilical and non-totally geodesic hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. In our 2010 paper we proved that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in an (n + 1)-dimensional sphere Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H and H3 are constant. In this paper, we prove the same results, without the assumption that H3 is constant. In fact, we show that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H is constant.

In this paper, we formulate the notion of the $\mathcal{F}$-stability of self-shrinking solutions to mean curvature flow in arbitrary codimension. Then we give some classifications of the $\mathcal{F}$-stable self-shrinkers in arbitrary codimension, in codimension one case, our results reduce to Colding-Minicozzi's results., 23 pages

We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space., 15 pages

In this paper, we use the inverse mean curvature flow to establish an optimal Minkowski type inequality, a weighted Alexandrov-Fenchel inequality for the mean convex star-shaped hypersurfaces in the Reissner-Nordstrom-anti-deSitter manifold, and a Penrose type inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over the Reissner-Nordstrom-anti-deSitter manifold.

Molecular adjuvants were used to augment the amplitude of the immune response in many studies recently. Ubiquitin (ub), the peptide binding truncated C-terminal portion of heat shock protein 70 (hsp70c) and interleukin-2 (IL-2) are widely investigated adjuvants which have been proved to be efficient. In our study, we compared the enhancing ability of these three adjuvants based on DNA vaccination using the porcine circovirus type 2 ORF2 (capsid) gene in mice. The results of lymphocyte proliferation assay, flow cytometric analysis (FCM), antibody titer and cytokine production showed that ub conjugated plasmid induced a stronger Th1 type cellular immune response and an observably higher level of Cap-specific serum immunoglobulin G antibody compared with hsp70c or IL-2 conjugated plasmids during the period of post-immunization. Meanwhile, the ub conjugation vaccinated group elicited stronger specific immunity against PCV2 challenge than the others during most of the time of post-challenge. Thus, these data indicate that ub is a superior adjuvant for a PCV2 DNA vaccination than the hsp70c and IL-2 molecules.