arXiv research

We complete a minor gap in Gromoll and Walschap classification of metric fibrations from the Euclidean space, thus completing the classification of Riemannian foliations on Euclidean spaces.

We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal resu…

Survey on complex submanifolds in Euclidean spaces.

The paper defines a metric on Euclidean triangles and polygons, proving properties and completeness.

The study classifies complete self-shrinkers in Euclidean space.

The purpose of this paper is to study complete $λ$-surfaces in Euclidean space $\mathbb R^3$. A complete classification for 2-dimensional complete $λ$-surfaces in Euclidean space $\mathbb R^3$ with constant squared norm of the second fundamental form is given.

By Hartman--Nirenberg's theorem, any complete flat hypersurface in Euclidean space must be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples. Flat fronts are flat hypersurfaces with admissible singularities. Murata--Umehara gave a representation formula for comp…

Study on minimal submanifolds with finite curvature in Euclidean space.

The purpose of this paper is to study complete self-shrinkers of mean curvature flow in Euclidean spaces. In the paper, we give a complete classification for 2-dimensional complete Lagrangian self-shrinkers in Euclidean space $\mathbb R^4$ with constant squared norm of the second fundamental form.

Prove Gromov's Euclidean endpoint $C^0$ rigidity conjecture for positive mass theorem.

The main purpose of this paper is to complete the work initiated by Sbrana in 1909 giving a complete local classification of the nonflat infinitesimally bendable hypersurfaces in Euclidean space.

CAT(0) spaces close to Euclidean spheres are homeomorphic to Euclidean spaces.

In this paper, the pinching problems of complete $λ$-hypersurfaces in a Euclidean space $\mathbb R^{n+1}$ are studied. By making use of the Sobolev inequality, we prove a global pinching theorem of complete $λ$-hypersurfaces in a Euclidean space $\mathbb R^{n+1}$.

CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.

CAT(0) spaces quasi-isometric to Euclidean spaces are homeomorphic to R^n.

Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold admits a flat, symmetric and complete connection. If the completeness assumption i…

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bi…

Researchers classify 3D self-shrinkers in 4D space.

Researchers describe a specific type of submanifolds in Euclidean space.

We show that a complete Euclidean submanifold with minimal index of relative nullity $ν_0>0$ and Ricci curvature with a certain controlled decay must be a $ν_0$-cylinder. This is an extension of the classical Hartman cylindricity theorem.

Flat subsets in Euclidean buildings are contained within apartments.

Study on Gauss images of specific minimal surfaces with finite curvature.

Study shows expanding Ricci solitons from specific metric cones.

The study provides bounds for geodesic diameter in Euclidean space.

We prove that strong finite total curvature complete hypersurfaces of (n+1)-euclidean space are proper and diffeomorphic to a compact manifold minus finitely many points. With an additional condition, we also prove that the Gauss map of such hypersurfaces extends continuously to the punctures.

In this paper, we introduce canonical principal direction (CPD) submanifolds with higher codimension in Euclidean spaces. We obtain the complete classification of surfaces endowed with CPD in the Euclidean 4-space.

We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also con…

Study classifies 3D self-shrinkers with constant second form norm.

In this note we provide a direct proof of the complete classification of conformally flat isoparametric submanifolds of Euclidean space.

The paper proves pseudolocality theorems for Ricci flows on incomplete manifolds.

In this paper, we give a complete description of all translation hypersurfaces with constant r-curvature Sr, in the Euclidean space.

We investigate complete minimal hypersurfaces in the Euclidean space $$, with Gauss-Kronecker curvature identically zero. We prove that, if $f:M^{3}\to {R}^{4}$ is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature b…

Study expanding gradient Ricci solitons with Euclidean base.

The paper classifies conformal solitons in pseudo-Euclidean spaces.

The uncertainty principle lemma for the Laplacian on Euclidean spaces shows the borderline-behavior of a potential for the following question : whether the Schrödinger operator has a finite or infinite number of the discrete pectrum. In this paper, we will give a generalization of this lemma on Euclidean spaces to that…

The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in Euclidean space. We prove (under certain technical…

The paper proves uniqueness of Ricci flow on noncompact manifolds.

We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface $M$ in $\mathbb{R}^{4}$ with zero scalar curvature $S_2$, nonzero Gauss-Kronecker…

The paper classifies invariant gradient $k$-Yamabe solitons in pseudo-Euclidean spaces.

Study on anisotropic curvature flow for noncompact convex hypersurfaces.

Constructs hyperspheres with prescribed mean curvature in Euclidean space.

The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

Survey on 4-manifolds with specific curvature properties.

New non-quadratic Euclidean complete affine maximal type hypersurfaces found for N≥2, θ∈(0,(N-1)/N].

Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.

Minimal isometric immersions F in codimension two from a complete Kahler manifold into Euclidean space had been classified for dimension greater than or equal to 3. In this note we describe the non--minimal situation by showing that, if F is real analytic but not everywhere minimal, then F is a cylinder over a real Kah…

The paper classifies a type of solitons in Euclidean spaces.

We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property of the Gauss image under the mean curvature flow we prove the long time existence…