Blowup behavior of harmonic maps with finite index

In this paper, we study the blow-up phenomena on the \(\alpha _k\)-harmonic map sequences with bounded uniformly \(\alpha _k\)-energy, denoted by \(\{u_{\alpha _k}: \alpha _k>1 \quad \text{ and } \quad \alpha _k\searrow 1\}\), from a compact Riemann surface into a compact Riemannian manifold. If the Ricci curvature of the target manifold has a positive lower bound and the indices of the \(\alpha _k\)-harmonic map sequence with respect to the corresponding \(\alpha _k\)-energy are bounded, then we can conclude that, if the blow-up phenomena occurs in the convergence of \(\{u_{\alpha _k}\}\) as \(\alpha _k\searrow 1\), the limiting necks of the convergence of the sequence consist of finite length geodesics, hence the energy identity holds true. For a harmonic map sequence \(u_k:(\Sigma ,h_k)\rightarrow N\), where the conformal class defined by \(h_k\) diverges, we also prove some similar results.     

First eigenvalues of geometric operators under the Yamabe flow

Suppose \((M,g_0)\) is a compact Riemannian manifold without boundary of dimension \(n\ge 3\). Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of \(g_0\) with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.     

$$L^p$$-Analysis of the Hodge–Dirac Operator Associated with Witten Laplacians on Complete Riemannian Manifolds

We prove R-bisectoriality and boundedness of the \(H^\infty \)-functional calculus in \(L^p\) for all \(1<p<\infty \) for the Hodge–Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry–Emery Ricci curvature on k-forms.     

On $$L^p$$-Boundedness of Pseudo-Differential Operators of Sjöstrand’s Class

We extended the known result that symbols from modulation spaces \(M^{\infty ,1}(\mathbb {R}^{2n})\), also known as the Sjöstrand’s class, produce bounded operators in \(L^2(\mathbb {R}^n)\), to general \(L^p\) boundedness at the cost of loss of derivatives. Indeed, we showed that pseudo-differential operators acting from \(L^p\)-Sobolev spaces \(L^p_s(\mathbb {R}^n)\) to \(L^p(\mathbb {R}^n)\) spaces with symbols from the modulation space \(M^{\infty ,1}(\mathbb {R}^{2n})\) are bounded, whenever \(s\ge n|1/p-1/2|.\) This estimate is sharp for all \(1< p<\infty \).     

On Submanifolds with 2-Type Pseudo-Hyperbolic Gauss Map in Pseudo-Hyperbolic Space

In this paper, we study pseudo-Riemannian submanifolds of a pseudo-hyperbolic space \(\mathbb H^{m-1}_s (-1) \subset \mathbb E^m_{s+1}\) with 2-type pseudo-hyperbolic Gauss map. We give a characterization of proper pseudo-Riemannian hypersurfaces in \(\mathbb H^{n+1}_s (-1) \subset \mathbb E^{n+2}_{s+1}\) with non-zero constant mean curvature and 2-type pseudo-hyperbolic Gauss map. For \(n=2\), we prove classification theorems. In addition, we show that the hyperbolic Veronese surface is the only maximal surface fully lying in \(\mathbb H^4_2 (-1) \subset \mathbb H^{m-1}_2 (-1)\) with 2-type pseudo-hyperbolic Gauss map. Moreover, we prove that a flat totally umbilical pseudo-Riemannian hypersurface \(M^n_t\) of the pseudo-hyperbolic space \(\mathbb {H}^{n+1}_t(-1) \subset \mathbb E^{n+2}_{t+1}\) has biharmonic pseudo-hyperbolic Gauss map.     

Existence of self-Cheeger sets on Riemannian manifolds

Let \(({\mathcal M},g)\) be a smooth compact Riemannian manifold of dimension \(N\ge 2\). We prove the existence of a family \((\Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) of self-Cheeger sets in \(({\mathcal M},g)\). The domains \(\Omega _\varepsilon \subset {\mathcal M}\) are perturbations of geodesic balls of radius \(\varepsilon \) centered at \(p \in {\mathcal M}\), and in particular, if \(p_0\) is a non-degenerate critical point of the scalar curvature of g, then the family \((\partial \Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) constitutes a smooth foliation of a neighborhood of \(p_0\).     

CMC Foliations of Closed Manifolds

We prove that every closed, smooth \(n\)-manifold \(X\) admits a Riemannian metric together with a constant mean curvature (CMC) foliation if and only if its Euler characteristic is zero, where by a CMC foliation we mean a smooth, codimension-one, transversely oriented foliation with leaves of CMC and where the value of the CMC can vary from leaf to leaf. Furthermore, we prove that this CMC foliation of \(X\) can be chosen so that when \(n\ge 2\), the constant values of the mean curvatures of its leaves change sign. We also prove a general structure theorem for any such non-minimal CMC foliation of \(X\) that describes relationships between the geometry and topology of the leaves, including the property that there exist compact leaves for every attained value of the mean curvature.     

Minimal Whitney stratification and Euler obstruction of discriminants

In this work we show how to obtain the minimal Whitney stratification of the discriminant of finitely determined map germs from \((\mathbb {C}^m,0)\) to \((\mathbb {C}^p,0)\), of corank one if \(n<p\), and only with \(A_k\) singularities, when \(m = n+p\) with \( n \ge 0\). We apply the theory developed by Gaffney which shows how to compute a Whitney stratification of discriminants of any finitely determined holomorphic map germ in the nice dimensions of Mather, or in its boundary. For the pairs cited above we show that both stratifications coincide. We also compute the local Euler obstruction at 0 in a class of discriminants of finitely determined map germs from \(\mathbb {C}^{n+p}\) to \(\mathbb {C}^p\) with \(n\ge 0\) and only with \(A_k\) singularities.     

Diameter growth and bounded topology of complete manifolds with nonnegative Ricci curvature

In this note, we show that a complete n-dim Riemannian manifold with nonnegative Ricci curvature is of finite topological type provided that the diameter growth of M is of order \(o(r^{((n-1)\alpha +1)/n})\) and the sectional curvature is no less than \(-{\frac{c}{r^{2\alpha }}}\) (here, \(0 \le \alpha \le 1\) and c is some positive constant) outside a geodesic ball large enough. In particular, if in a neighborhood of an isolated end of the manifold in question, the above assumptions are satisfied, then the end has a collared neighborhood.     

Elimination of extremal index zeroes from generic paths of closed $$$$-forms

Let \( \alpha \) be a Morse closed \( 1 \)-form of a smooth \( n \)-dimensional manifold \( M \). The zeroes of \( \alpha \) of index \( 0 \) or \( n \) are called centers. It is known that every non-vanishing de Rham cohomology class \( u \) contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the last statement: every generic path \( (\alpha _t)_{ t\in [0,1] }\) of closed \( 1 \)-forms in a fixed class \( u\ne 0 \) such that \( \alpha _0,\alpha _1 \) have no centers, can be modified relatively to its extremities to another such path \( (\beta _t)_{t \in [0,1]} \) having no center at all.     

On generalized Toeplitz and little Hankel operators on Bergman spaces

We find a concrete integral formula for the class of generalized Toeplitz operators \(T_a\) in Bergman spaces \(A^p\), \(1<p<\infty \), studied in an earlier work by the authors. The result is extended to little Hankel operators. We give an example of an \(L^2\)-symbol a such that \(T_{|a|} \) fails to be bounded in \(A^2\), although \(T_a : A^2 \rightarrow A^2\) is seen to be bounded by using the generalized definition. We also confirm that the generalized definition coincides with the classical one whenever the latter makes sense.     

Remarks on the nonexistence of biharmonic maps

In this short note we study a nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that \({\phi : (M, g) \to (N, h)}\) is a biharmonic map, where (M, g) is a complete Riemannian manifold and (N, h) a Riemannian manifold with nonpositive sectional curvature, we will prove that \({\phi}\) is a harmonic map if one of the following conditions holds: (i) \({|d\phi|}\) is bounded in L^q(M) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 \leq q \leq \infty}\), \({1 < p < \infty}\); or (ii) \({Vol(M) = \infty}\) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 < p < \infty}\). In addition, if N has strictly negative sectional curvature, we assume that \({rank\phi(q) \geq 2}\) for some \({q \in M}\) and \({\int_{M}|\tau(\phi)|^{p}dv_{g} < \infty}\), for some \({1 < p < \infty}\). These results improve the related theorems due to Baird et al. (cf. Ann Golb Anal Geom 34:403–414, 2008), Nakauchi et al. (cf. Geom. Dedicata 164:263–272, 2014), Maeta (cf. Ann Glob Anal Geom 46:75–85, 2014), and Luo (cf. J Geom Anal 25:2436–2449, 2015).     

Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

For a sequence of approximate harmonic maps \((u_n,v_n)\) (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form \(g_N -\beta dt^2\) for some Riemannian metric \(g_N\) and some positive function \(\beta \) on N, we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.     

On the approximation of dynamical indicators in systems with nonuniformly hyperbolic behavior

Let f be a \(C^{1+\alpha }\) diffeomorphism of a compact Riemannian manifold and \(\mu \) an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that the topological pressure \(P(f|\Omega _n,\phi )\) converges to the free energy \(P_{\mu }(\phi ) = h(\mu ) + \int \phi {d\mu }\). We also prove that for a suitable class of potentials \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that \(P(f|\Omega _n,\phi ) \rightarrow P(\phi )\).     

Rigidity of complete manifolds with parallel Cotton tensor

The aim of this paper is to show some rigidity results for complete Riemannian manifolds with parallel Cotton tensor. In particular, we prove that any compact manifold of dimension \(n\ge 3\) with parallel Cotton tensor and positive constant scalar curvature is isometric to a finite quotient of \({\mathbb {S}}^n\) under a pointwise or integral pinching condition. Moreover, a rigidity theorem for stochastically complete manifolds with parallel Cotton tensor is also given. The proofs rely mainly on curvature elliptic estimates and the weak maximum principle.     

The $$$$-Affine Caacity

In this paper, the \(p\)-affine capacity is introduced for \(1<p<n\) and then developed to discover the upper and lower isocapacitary inequalities that strengthen optimally both the Maz’ya \(p\)-isocapacitary inequality and the Lutwak–Yang–Zhang \(L_p\) affine isoperimetric inequality over the \(n\)-dimensional Euclidean space \({\mathbb {R}}^{n}\).     

Large conformal metrics with prescribed sign-changing Gauss curvature

Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that

$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$

Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies

$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$

and

$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$

where \(\delta _{p}\) designates Dirac mass at the point \(p\).

     

Ricci Flow on a Class of Noncompact Warped Product Manifolds

We consider the Ricci flow on noncompact \(n+1\)-dimensional manifolds M with symmetries, corresponding to warped product manifolds \(\mathbb {R}\times T^n\) with flat fibres. We show longtime existence and that the Ricci flow solution is of type III, i.e. the curvature estimate \(|{{\mathrm{Rm}}}|(p,t) \le C/t\) for some \(C > 0\) and all \(p \in M, t \in (1,\infty )\) holds. We also show that if M has finite volume, the solution collapses, i.e. the injectivity radius converges uniformly to 0 (as \(t \rightarrow \infty \)) while the curvatures stay uniformly bounded, and furthermore, the solution converges to a lower dimensional manifold. Moreover, if the (n-dimensional) volumes of hypersurfaces coming from the symmetries of M are uniformly bounded, the solution converges locally uniformly to a flat cylinder after appropriate rescaling and pullback by a family of diffeomorphisms. Corresponding results are also shown for the normalized (i.e. volume preserving) Ricci flow.     

<Emphasis Type="Italic">p</Emphasis>-Fundamental tone estimates of submanifolds with bounded mean curvature

In this paper, we estimate the p-fundamental tone of submanifolds in a Cartan–Hadamard manifold. First, we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension-one \(C^2\)-foliations of open subsets \(\Omega \) of Riemannian manifolds M and obtain lower bound estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of \(\Omega \).