Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds

In this paper we study gradient estimates for the positive solutions of the porous medium equation: $$u_t=\Delta u^m$$ where m>1, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li–Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results improve the ones of Lu, Ni, Vázquez, and Villani (in J. Math. Pures Appl. 91:1–19, 2009). Moreover, our results recover the ones of Davies (in Cambridge Tracts Math vol. 92, 1989), Hamilton (in Comm. Anal. Geom. 1:113–125, 1993) and Li and Xu (in Adv. Math. 226:4456–4491, 2011).     

Integral Ricci curvature bounds along geodesics for nonexpanding gradient Ricci solitons

Following Li and Yau (Acta Math 156:153?C201 1986) and similar to Perelman (The entropy formula for the Ricci flow and its geometric applications), we define an energy functional ${\mathcal{J}}$ associated to a smooth function ${\phi}$ on a complete Riemannian manifold. As an application, we deduce integral Ricci curvature upper bounds along modified geodesics for complete steady and shrinking gradient Ricci solitons.     

On rigidity of hypersurfaces with constant curvature functions in warped product manifolds

In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as “weighted” mean curvatures, which extend the work (Brendle in Publ Math Inst Hautes Études Sci 117:247–269, 2013; Brendle and Eichmair in J Differ Geom 94(94):387–407, 2013; Montiel in Indiana Univ Math J 48:711–748, 1999) considering constant mean curvature functions. Secondly, we obtain the rigidity results for hypersurfaces in the space forms with constant linear combinations of intrinsic Gauss–Bonnet curvatures $L_k$ . To achieve this, we develop some new kind of Newton–Maclaurin type inequalities on $L_k$ which may have independent interest.     

Bach-flat gradient steady Ricci solitons

In this paper we prove that any n-dimensional (n ≥ 4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in Cao and Chen (Trans Am Math Soc 364:2377–2391, 2012) and Catino and Mantegazza (Ann Inst Fourier 61(4):1407–1435, 2011).     

Divergence points and normal numbers

In Olsen and Winter (J Lond Math Soc 67(2):103–122, 2003) and Baek et?al. (Advan Math 214:267–287, 2007) the authors have introduced the notion of “normal” and “non-normal” points of a self-similar set as a main tool for studying the Hausdorff and the packing dimensions of a set of divergence points of self-similar measures. In this paper we will extend the results about the Hausdorff and the packing dimensions of “non-normal” points of a self-similar set in a point of view of Bisbas (Bulletin des Sciences Mathématiques 129(1):25–37, 2005). Namely, we will prove that both the Hausdorff and packing dimensions remain the same if we consider subsets determined by the normality to some bases. This will be proved using the techniques from Bisbas (Bulletin des Sciences Mathématiques 129(1):25–37, 2005) and the construction of suitable measures. Simultaneously this will also give simpler proofs of some of the results from Olsen and Winter (J Lond Math Soc 67(2):103–122, 2003) and Baek et?al. (Advan Math 214:267–287, 2007).     

Normalized Ricci flows and conformally compact Einstein metrics

In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles to establish the regularity of conformal compactness along the normalized Ricci flow including that of the limit metric at time infinity. Therefore we are able to recover the existence results in Graham and Lee (Adv Math 87:186–255, 1991), Lee (Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds, 2006), and Biquard (Surveys in Differential Geometry: Essays on Einstein Manifolds, 1999) of conformally compact Einstein metrics with conformal infinities which are perturbations of that of given non-degenerate conformally compact Einstein metrics.     

A Note on Aubin-Lions-Dubinskiĭ Lemmas

Strong compactness results for families of functions in seminormed nonnegative cones in the spirit of the Aubin-Lions-Dubinski? lemma are proven, refining some recent results in the literature. The first theorem sharpens slightly a result of Dubinski? (in Mat. Sb. 67(109):609–642, 1965) for seminormed cones. The second theorem applies to piecewise constant functions in time and sharpens slightly the results of Dreher and Jüngel (in Nonlinear Anal. 75:3072–3077, 2012) and Chen and Liu (in Appl. Math. Lett. 25:2252–2257, 2012). An application is given, which is useful in the study of porous-medium or fast-diffusion type equations.     

Index sets and Scott sentences

For a computable structure \({\mathcal{A}}\) , there may not be a computable infinitary Scott sentence. When there is a computable infinitary Scott sentence \({\varphi}\) , then the complexity of the index set \({I(\mathcal{A})}\) is bounded by that of \({\varphi}\) . There are results (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012; Knight and Saraph in Scott sentences for certain groups, pre-print) giving “optimal” Scott sentences for structures of various familiar kinds. These results have been driven by the thesis that the complexity of the index set should match that of an optimal Scott sentence (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012). In this note, it is shown that the thesis does not always hold. For a certain subgroup of \({\mathbb{Q}}\) , there is no computable d- \({\Sigma_2}\) Scott sentence, even though (as shown in Ash and Knight in Scott sentences for certain groups, pre-print) the index set is d- \({\Sigma^0_2}\) .     

A remark on the paper “Laterally closed lattice homomorphisms”

A new and simple proof of the main result of the paper “Laterally closed lattice homomorphisms” by Toumi and Toumi (J Math Anal Appl 324:1178–1194, 2006) is given following the paper “Extension of Riesz homomorphisms, I” by Buskes (J Aust Math Soc Ser A 39(1):107–120, 1985).     

On the rigidity of hypersurfaces into space forms

Our purpose is to study the rigidity of complete hypersurfaces immersed into a Riemannian space form. In this setting, first we use a classical characterization of the Euclidean sphere \(\mathbb S ^{n+1}\) due to Obata (J Math Soc Jpn 14:333–340, 1962) in order to prove that a closed orientable hypersurface \(\Sigma ^n\) immersed with null second-order mean curvature in \(\mathbb S ^{n+1}\) must be isometric to a totally geodesic sphere \(\mathbb S ^{n}\) , provided that its Gauss mapping is contained in a closed hemisphere. Furthermore, as suitable applications of a maximum principle at the infinity for complete noncompact Riemannian manifolds due to Yau (Indiana Univ Math J 25:659–670, 1976), we establish new characterizations of totally geodesic hypersurfaces in the Euclidean and hyperbolic spaces. We also obtain a lower estimate of the index of minimum relative nullity concerning complete noncompact hypersurfaces immersed in such ambient spaces.     

The canonical shrinking soliton associated to a Ricci flow

To every Ricci flow on a manifold ${\mathcal{M}}$ over a time interval ${I\subset\mathbb{R}_-}$ , we associate a shrinking Ricci soliton on the space–time ${\mathcal{M}\times I}$ . We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author Topping (J Reine Angew Math 636:93–122, 2009), and McCann and the second author (Am J Math 132:711–730, 2010); we briefly survey the link between these subjects.     

Tilting Theory and Functor Categories I. Classical Tilting

Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel (1988) and Cline et al. (J Algebra 304:397–409 1986) proved that generalized tilting induces derived equivalences between module categories, and tilting complexes were used by Rickard (J Lond Math Soc 39:436–456, 1989) to develop a general Morita theory of derived categories. On the other hand, functor categories were introduced in representation theory by Auslander (I Commun Algebra 1(3):177–268, 1974), Auslander (1971) and used in his proof of the first Brauer–Thrall conjecture (Auslander 1978) and later on, used systematically in his joint work with I. Reiten on stable equivalence (Auslander and Reiten, Adv Math 12(3):306–366, 1974), Auslander and Reiten (1973) and many other applications. Recently, functor categories were used in Martínez-Villa and Solberg (J Algebra 323(5):1369–1407, 2010) to study the Auslander–Reiten components of finite dimensional algebras. The aim of this paper is to extend tilting theory to arbitrary functor categories, having in mind applications to the functor category Mod (mod_Λ), with Λ a finite dimensional algebra.     

Rigidity of manifolds with Bakry–émery Ricci curvature bounded below

Let M be a complete Riemannian manifold with Riemannian volume vol _g and f be a smooth function on M. A sharp upper bound estimate on the first eigenvalue of symmetric diffusion operator ${\Delta_f = \Delta- \nabla f \cdot \nabla}$ was given by Wu (J Math Anal Appl 361:10?C18, 2010) and Wang (Ann Glob Anal Geom 37:393?C402, 2010) under a condition that finite dimensional Bakry?Cémery Ricci curvature is bounded below, independently. They propounded an open problem is whether there is some rigidity on the estimate. In this note, we will solve this problem to obtain a splitting type theorem, which generalizes Li?CWang??s result in Wang (J Differ Geom 58:501?C534, 2001, J Differ Geom 62:143?C162, 2002). For the case that infinite dimensional Bakry?CEmery Ricci curvature of M is bounded below, we do not expect any upper bound estimate on the first eigenvalue of ?? _f without any additional assumption (see the example in Sect. 2). In this case, we will give a sharp upper bound estimate on the first eigenvalue of ?? _f under the additional assuption that ${|\nabla f|}$ is bounded. We also obtain the rigidity result on this estimate, as another Li?CWang type splitting theorem.     

The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups

The general surface group conjecture asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in Fine et al. (Sci Math A 1:1–15, 2008). First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if G is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then G is free, a surface group or a solvable Baumslag–Solitar Group. Further combining results in Fine et al. (Sci Math A 1:1–15, 2008) on Property IF with a theorem of Wilton (Geom Topol, 2012) and results of Stallings (Ann Math 2(88):312–334, 1968) and Kharlampovich and Myasnikov (Trans Am Math Soc 350(2):571–613, 1998) we show that Surface Group Conjecture C proposed in Fine et al. (Sci Math A 1:1–15, 2008) is true, namely: If G is a finitely generated nonfree freely indecomposable fully residually free group with property IF, then G is a surface group.     

The Local Langlands Correspondence for GL n over p-adic fields

We extend our methods from Scholze (Invent. Math. 2012, doi:10.1007/s00222-012-0419-y) to reprove the Local Langlands Correspondence for GL _n over p-adic fields as well as the existence of ?-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001). In contrast to the proofs of the Local Langlands Correspondence given by Henniart (Invent. Math. 139(2), 439–455, 2000), and Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001), our proof completely by-passes the numerical Local Langlands Correspondence of Henniart (Ann. Sci. Éc. Norm. Super. 21(4), 497–544, 1988). Instead, we make use of a previous result from Scholze (Invent. Math. 2012, doi:10.1007/s00222-012-0419-y) describing the inertia-invariant nearby cycles in certain regular situations.     

Some Recent Developments on Hopf��s Holomorphic Form

Hopf??s theorem on surfaces in ${\mathbb{R}^3}$ with constant mean curvature (Hopf in Math Nach 4:232?C249, 1950-51) was a turning point in the study of such surfaces. In recent years, Hopf-type theorems appeared in various ambient spaces, (Abresch and Rosenberg in Acta Math 193:141?C174, 2004 and Abresch and Rosenberg in Mat Contemp Sociedade Bras Mat 28:283-298, 2005). The simplest case is the study of surfaces with parallel mean curvature vector in ${M_k^n \times \mathbb{R}, n \ge 2}$ , where ${M_k^n}$ is a complete, simply-connected Riemannian manifold with constant sectional curvature k ?? 0. The case n?=?2 was solved in Abresch and Rosenberg 2004. Here we describe some new results for arbitrary n.     

Sharp spectral gap and Li–Yau’s estimate on Alexandrov spaces

In the previous work (Zhang and Zhu in J Differ Geom, http://arxiv.org/pdf/1012.4233v3, 2012), the second and third authors established a Bochner type formula on Alexandrov spaces. The purpose of this paper is to give some applications of the Bochner type formula. Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen and Wang (Sci Sin (A) 37:1–14, 1994), Chen and Wang (Sci Sin (A) 40:384–394, 1997) and Bakry–Qian (Adv Math 155:98–153, 2000), from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. Secondly, we obtain (sharp) Li–Yau’s estimate for positve solutions of heat equations on Alexandrov spaces.     

A note on K?hler–Ricci flow

Let g(t) with ${t\in [0,T)}$ be a complete solution to the K?hler–Ricci flow: ${\frac{d}{dt}g_{i\bar j}=-R_{i\bar j}}$ where T may be ∞. In this article, we show that the curvature of g(t) is uniformly bounded if the solution g(t) is uniformly equivalent. This result is stronger than the main result in ?e?um (Am J Math 127(6):1315–1324, 2005) within the category of K?hler–Ricci flow.     

L p -theory for second-order elliptic operators with unbounded coefficients

Second-order elliptic operators with unbounded coefficients of the form ${Au := -{\rm div}(a\nabla u) + F . \nabla u + Vu}$ in ${L^{p}(\mathbb{R}^{N}) (N \in \mathbb{N}, 1 < p < \infty)}$ are considered, which are the same as in recent papers Metafune et?al. (Z Anal Anwendungen 24:497–521, 2005), Arendt et?al. (J Operator Theory 55:185–211, 2006; J Math Anal Appl 338: 505–517, 2008) and Metafune et?al. (Forum Math 22:583–601, 2010). A new criterion for the m-accretivity and m-sectoriality of A in ${L^{p}(\mathbb{R}^{N})}$ is presented via a certain identity that behaves like a sesquilinear form over L ^p ×?L ^p'. It partially improves the results in (Metafune et?al. in Z Anal Anwendungen 24:497–521, 2005) and (Metafune et?al. in Forum Math 22:583–601, 2010) with a different approach. The result naturally extends Kato’s criterion in (Kato in Math Stud 55:253–266, 1981) for the nonnegative selfadjointness to the case of p ≠?2. The simplicity is illustrated with the typical example ${Au = -u\hspace{1pt}'' + x^{3}u\hspace{1pt}' + c |x|^{\gamma}u}$ in ${L^p(\mathbb{R})}$ which is dealt with in (Arendt et?al. in J Operator Theory 55:185–211, 2006; Arendt et?al. in J Math Anal Appl 338: 505–517, 2008).