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Since $n$-dimensional $\lambda$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete $\lambda$-hypersurfaces. We give a gap theorem of complete $\lambda$-hypersurfaces with polynomial area growth. By making use of the generalized maximum principle for $\mathcal L$ of $\lambda$-hypersurfaces, we prove a rigidity theorem of complete $\lambda$-hypersurfaces., Comment: Comments are welcome

In this paper, we study a kind of geometrically constrained variational problem of the Willmore functional. A surface l : Σ → C is called a Lagrangian–Willmore surface (in short, a LW surface) or a Hamiltonian–Willmore surface (in short, a HW surface) if it is a critical point of the Willmore functional under Lagrangian deformations or Hamiltonian deformations, respectively. We extend the L∞ estimates of the second fundamental form of Willmore surfaces to both HW and LW surfaces and thus get a gap theorem for both HW and LW surfaces. To investigate the existence of HW surfaces we introduce a sixth-order flow which is called by us the Hamiltonian–Willmore flow (in short, the HW flow) decreasing the Willmore energy and we prove that this flow is well posed.

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based on continuous rescaling and the classification of self-shrinkers. We show that all notions of singular sets defined in \cite{St} coincide for any Type I mean curvature flow, thus generalizing the result of Stone who established that for any mean convex Type I Mean curvature flow. We also establish a gap theorem for self-shrinkers., Comment: 21 pages

In this paper, we show that any ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a generalization of Anderson's result for Ricci-flat manifolds. As a corollary, a gap theorem for gradient shrinking Ricci solitons is also obtained.