[29] A. Georgakopoulos, M. Magliaro, L. Mari & A. Savas-Halilaj
Title: Harmonic maps from S3 to S2 and the rigidity of the Hopf fibration
Abstract: It was conjectured by Eells that the only harmonic maps from S3 to S2 are Hopf fibrations composed with conformal maps of S2. We support this conjecture by proving its validity under suitable conditions on the Hessian and the singular values of the map. Among the results, we obtain a pinching theorem in the spirit of that of Simons, Lawson, and Chern, do Carmo & Kobayashi for minimal hypersurfaces in the sphere.
Journal: arXiv 2511.16522 (2025), 1-21.
[28] G. Habib & A. Savas-Halilaj
Title: Harmonic unit vector fields on 3-manifolds
Abstract: We investigate harmonic unit vector fields with totally geodesic integral curves on 3-manifolds. Under mild curvature assumptions, we classify both the vector fields and the manifolds that support them. Our results are inspired by Carriere's classfication of Riemannian flows on compact three manifolds, as well as by the works of Geiges and Belgun on Sasakian manifolds.
Journal: arXiv 2510.19756 (2025), 1-21.
[27] M. Magliaro, L. Mari, F. Roing & A. Savas-Halilaj
Title: On mean curvature flow solitons in the sphere
Abstract: In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere moving along the integral curves of the Hopf vector field. While such solitons must necessarily be minimal if compact, we a produce non-minimal, complete cylindrical example which wraps around the Clifford torus along each end, it has reflection and rotational symmetry and its mean curvature changes sign on each end. Indeed, we prove that a complete 2-dimensional soliton with non-negative mean curvature outside a compact set must be a covering of a Clifford torus. Concluding, we obtain a pinching theorem under suitable conditions on the second fundamental form.
Journal: arXiv: 2502.09199 (2025), 1-24.
[26] A. Savas-Halilaj & K. Smoczyk
Title: Codimension two mean curvature flow of entire graphs
Abstract: We consider the graphical mean curvature flow of maps f:Rm->Rn, m>1, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well-known maximum principle of Ecker and Huisken. In the case of uniformly area decreasing maps f:Rm->R2, m>1, we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self-expander.
Journal: J. Lond. Math. Soc. 110 (2024), 1-33.
[25] M. Magliaro, L. Mari, F. Roing & A. Savas-Halilaj
Title: Sharp pinching theorems for complete submanifolds in the sphere
Abstract: We prove that every complete, minimally immersed hypersurface M in the unit sphere whose second fundamental form is less or equal than the dimension of the hypersurface is either totally geodesic or a Riemannian covering of a minimal Clifford torus; thereby extending Simons’ pinching theorem from compact to the complete category. We also extend the corresponding results for compact hypersurfaces with non-vanishing constant mean curvature, due to Alencar & do Carmo, to complete immersed CMC hypersurfaces, under the optimal bound for their umbilicity tensor. In dimension n<7, a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work of Santos. Our approach is inspired by the conformal method due to Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni.
Journal: J. Reine Angew. Math. (Crelle’s Journal) 814 (2024), 117-134.
[24] N. Roidos & A. Savas-Halilaj
Title: Curve shortening flow on Riemann surfaces with conical singularities
Abstract: In this paper we study curve shortening flow on Riemann surfaces with singular metrics. It turns out that this flow is governed by a degenerate quasilinear parabolic partial differential equation. Under natural geometric assumptions we prove short-time existence, uniqueness, and regularity of the flow. We also show that the evolving curves stay fixed at the singular points of the surface and derive collapsing and non-collapsing results.
Journal: Math. Ann. 390 (2024), 2337-2411.
[23] L. Mari, J. Rocha de Oliveira, A. Savas-Halilaj & R. Sodre de Sena
Title: Conformal solitons for the MCF in hyperbolic space
Abstract: In this paper we study conformal solitons for the mean curvature flow in hyperbolic space. Working in the upper half-space model, we focus on hοro-expanders. We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability, and address the case of noncompact boundaries contained between two parallel hyperplanes of the boundary at infinity of the hyperbolic space. We conclude by proving rigidity results for bowl and grim-reaper cylinders.
Journal: Ann. Glob. Anal. Geom. 65 (2024), Article No: 19.
[22] I. Fourtzis, M. Markellos & A. Savas-Halilaj
Title: Gauss maps of harmonic and minimal great circle fibrations
Abstract: We investigate Gauss maps associated to great circle fibrations of S3. We show that the Gauss map associated to such a fibration is harmonic, respectively minimal, if and only if the unit vector field generating the great circle foliation is harmonic, respectively minimal. These results can be viewed as analogues of the classical theorem of Ruh and Vilms about the harmonicity of the Gauss map of a minimal submanifold in the euclidean space. Moreover, we prove that a harmonic or minimal unit vector field in S3 with great circle integral curves is a Hopf vector field.
Journal: Ann. Glob. Anal. Geom. 63 (2023), Article No: 12.
[21] R. Assimos, A. Savas-Halilaj & K. Smoczyk
Title: Graphical mean curvature flow with bounded bi-Ricci curvature
Abstract: We consider the graphical mean curvature flow of a strictly area decreasing map between a compact Riemannian manifold M of dimension greater or equal than 2 and a complete Riemann surface N of bounded geometry. We prove long-time existence of the flow and that strictly area decreasing property is preserved, when the bi-Ricci curvature BRic of M is bounded from below by the sectional curvature K of N. In addition, we obtain smooth convergence to a minimal map if Ric ≥ sup{0,supK}. These results significantly improve existing results on the graphical mean curvature flow in codimension 2.
Journal: Calc. Var. Partial Differential Equations 62 (2023), Article No: 12.
[20] M. Markellos & A. Savas-Halilaj
Title: Rigidity of the Hopf fibration
Abstract: In this paper, we study minimal maps between euclidean spheres. The Hopf fibrations provide explicit examples of such minimal Riemannian submersions. Moreover, their corresponding graphs have second fundamental form of constant norm. We prove that a minimal submersion from S3 to S2 whose Gauss map satisfies a suitable pinching condition must be weakly conformal and with totally geodesic fibers. As a consequence, we obtain that an equivariant minimal submersion from S3 to S2 coincides with the Hopf fibration. Furthermore, we show that a minimal map from S3 to S2 with constant singular values and constant norm of the second fundamental form is either constant or coincides with the Hopf fibration.
Journal: Calc. Var. Partial Differential Equations 60 (2021), Article No: 171.
[19] A. Savas-Halilaj
Title: Graphical mean curvature flow
Abstract: In this survey article we discuss recent developments on the mean curvature flow of graphical submanifolds, generated by smooth maps between Riemannian or Kaehlerian manifolds. We discuss interesting applications of this technique, related with the homotopy type of smooth maps with small dilation between Riemannian manifolds. We treat also other topics as Bernstein type theorems for minimal graphs. The material of this paper is partly based on a series of lectures delivered by the author at the Chern Institute of Mathematics held in Tianjin-China in November 2019.
Book chapter: Nonlinear Analysis, Partial Differential Equations & Applications (2021), 1-80, Springer-Verlag.
[18] A. Savas-Halilaj
Title: A Schwarz-Pick lemma for minimal maps
Abstract: In this note, we prove a Schwarz-Pick type lemma for minimal maps between complete negatively curved Riemann surfaces. In fact, we prove that if f:M—>N is a minimal map with bounded Jacobian determinant between two complete negatively curved Riemann surfaces M and N whose sectional curvatures sec(M) and sec(N) satisfy infsec(M) ≥ supsec(N), then f is area decreasing.
Journal: Ann. Glob. Anal. Geom. 56 (2019), 193-201.
[17] A. Savas-Halilaj & K. Smoczyk
Title: Lagrangian mean curvature flow of Whitney spheres
Abstract: It is shown that an equivariant Lagrangian sphere with a positivity condition on its Ricci curvature develops a type-II singularity under the Lagrangian mean curvature flow that rescales to the product of a grim reaper with a flat Lagrangian space. In particular this result applies to the Whitney sphere.
Journal: Geometry & Topology 23 (2019), 1057-1084.
[16] F. Martin, J. Perez-Garcia, A. Savas-Halilaj & K. Smoczyk
Title: A characterization of the grim reaper cylinder
Abstract: In this article, we prove that a connected and properly embedded translating soliton in R3 with uniformly bounded genus on compact sets which is C1-asymptotic to two planes outside a cylinder, either is flat or coincide with the grim reaper cylinder.
Journal: J. Reine Angew. Math. (Crelle’s Journal) 746 (2019), 209-234.
[15] M. Dajczer, Th. Kasioumis, A. Savas-Halilaj & Th. Vlachos
Title: Complete minimal submanifolds with nullity in the hyperbolic space
Abstract: We investigate 3-dimensional complete minimal submanifolds in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in previous papers by us. If the scalar curvature is bounded from below we conclude that the submanifold has to be either totally geodesic or a generalised cone over a complete minimal surface lying in an equidistant submanifold of the hyperbolic space.
Journal: J. Geom. Anal. 29 (2019), 413-427.
[14] M. Dajczer, Th. Kasioumis, A. Savas-Halilaj & Th. Vlachos
Title: Complete minimal submanifolds with nullity in Euclidean spheres
Abstract: In this paper we investigate m-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least m−2 at any point. These are austere submanifolds in the sense of Harvey and Lawson and were initially studied by Bryant. For any dimension and codimension there is an abundance of non-complete examples fully described by Dajczer and Florit in terms of a class of surfaces, called elliptic, for which the ellipse of curvature of a certain order is a circle at any point. Under the assumption of completeness, it turns out that any submanifold is either totally geodesic or has dimension three. In the latter case there are plenty of examples, even compact ones. Under the mild assumption that the Omori-Yau maximum principle holds on the manifold, a trivial condition in the compact case, we provide a complete local parametric description of the submanifolds in terms of 1-isotropic surfaces in Euclidean space. These are the minimal surfaces for which the standard ellipse of curvature is a circle at any point. For these surfaces, there exists a Weierstrass type representation that generates all simply-connected ones.
Journal: Comment. Math. Helv. 93 (2018), 645-660.
[13] A. Savas-Halilaj & K. Smoczyk
Title: Mean curvature flow of area decreasing maps between surfaces
Abstract: In this article, we give a complete description of the evolution of an area decreasing map f : M → N induced by its mean curvature in the situation where M and N are complete Riemann surfaces with bounded geometry, M being compact, for which their sectional curvatures sec(M) and sec(N) satisfy min sec(M) ≥ sup sec(N).
Journal: Ann. Glob. Anal. Geom. 53 (2018), 11-37.
[12] M. Dajczer, Th. Kasioumis, A. Savas-Halilaj & Th. Vlachos
Title: Complete minimal submanifolds with nullity in Euclidean space
Abstract: In this paper, we investigate complete minimal submanifolds in Euclidean space with positive index of relative nullity. Let M be a complete m-dimensional Riemannian manifold and let f be an isometric minimal immersion of M to a euclidean space with relative nullity at least m-2. If the Omori-Yau maximum principle for the Laplacian holds on M, for instance if the scalar curvature on M does not decrease to minus infinity too fast or if f is proper, then the submanifold is just a cylinder over a minimal surface.
Journal: Math. Z. 287 (2017), 481-491.
[11] F. Martin, A. Savas-Halilaj & K. Smoczyk
Title: On the topology of translating solitons of the mean curvature flow
Abstract: In the present article, we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow in euclidean space.
Journal: Calc. Var. Partial Differential Equations 54 (2015), 2853-2882.
[10] A. Savas-Halilaj & K. Smoczyk
Title: Evolution of contractions by mean curvature flow
Abstract: In this article, we investigate length decreasing maps f between Riemannian manifolds M and N where M is compact of dimension greater than 1 and N is complete with bounded geometry. Under suitable and natural conditions on the curvatures of M and N we show that the mean curvature flow provides a smooth homotopy of f into a constant map.
Journal: Math. Ann. 361 (2015), 725-740.
[09] A. Savas-Halilaj & K. Smoczyk
Title: Homotopy of area decreasing maps by mean curvature flow
Abstract: Let f be a smooth area decreasing map between two compact Riemannian manifolds M and N. Under weak and natural assumptions on the curvatures of M and N, we prove that the mean curvature flow provides a smooth homotopy of the map f to a constant map.
Journal: Adv. Math. 255 (2014), 455-473.
[08] A. Savas-Halilaj & K. Smoczyk
Title: Bernstein theorems for length and area decreasing minimal maps
Abstract: In this paper we prove Liouville and Bernstein theorems in higher codimension for length and area decreasing maps between two Riemannian manifolds. The proofs are based on a strong elliptic maximum principle for sections in vector bundles, which we also present in this article.
Journal: Calc. Var. Partial Differential Equations 50 (2014), 549-577.
[07] A. Savas-Halilaj
Title: On deformable minimal hypersurfaces in space forms
Abstract: The aim of this paper is to complete the local classification of minimal hypersurfaces with vanishing Gauss-Kronecker curvature in a 4-dimensional space form. Moreover, we give a classification of complete minimal hypersurfaces with vanishing Gauss-Kronecker curvature and scalar curvature bounded from below.
Journal: J. Geom. Anal. 23 (2013), 1032-1057.
[06] Th. Hasanis, A. Savas-Halilaj & Th. Vlachos
Title: On the Jacobian of minimal graphs in R4
Abstract: We provide a characterization for complex analytic curves among two-dimensional minimal graphs in R4 via the Jacobian.
Journal: Bull. Lond. Math. Soc. 43 (2011), 321-327.
[05] Th. Hasanis, A. Savas-Halilaj & Th. Vlachos
Title: Minimal graphs with bounded Jacobians
Abstract: We obtain a Bernstein type result for entire two-dimensional minimal graphs in R4, which extends a previous result due to L. Ni. Moreover, we provide a characterization for complex analytic curves.
Journal: Proc. Amer. Math. Soc. 137 (2009), 3463-3471.
[04] Th. Hasanis, A. Savas-Halilaj & Th. Vlachos
Title: Complete minimal hypersurfaces in the hyperbolic space H4 with zero Gauss-Kronecker curvature
Abstract: We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space H4 with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, a nowhere vanishing second fundamental form and a scalar curvature bounded from below.
Journal: Trans. Amer. Math. Soc. 359 (2007), 2799-2818.
[03] Th. Hasanis, A. Savas-Halilaj & Th. Vlachos
Title: Minimal hypersurfaces of S4 with vanishing Gauss-Kronecker curvature
Abstract: In this paper, we investigate the structure of 3-dimensional complete minimal hypersurfaces in the unit sphere with Gauss-Kronecker curvature identically zero.
Journal: Math. Proc. Cambridge Philos. Soc. 142 (2007), 125-132.
[02] Th. Hasanis, A. Savas-Halilaj & Th. Vlachos
Title: Minimal hypersurfaces with zero Gauss-Kronecker curvature
Abstract: We investigate complete minimal hypersurfaces in the Euclidean space R4, with Gauss-Kronecker curvature identically zero. We prove that, if M is a complete minimal hypersurface lying in R4 with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below, then M splits as a Euclidean product L×R, where L is a complete minimal surface in R3 with Gaussian curvature bounded from below.
Journal: Illinois J. Math. 49 (2005), 523-529.
[01] Th. Hasanis, A. Savas-Halilaj & Th. Vlachos
Title: Complete minimal hypersurfaces in a sphere
Abstract: In this paper, we investigate complete minimal hypersurfaces M in unit euclidean spheres with at most two principal curvatures. We prove that if the squared norm S of the second fundamental form of M is greater or equal than n, then S=n and M is a minimal Clifford torus.
Journal: Monatsh. Math. 145 (2005), 301-307.
[MSc Thesis] A. Savas-Halilaj
Title: Hypersurfaces of the sphere with constant scalar curvature
University of Ioannina, Greece, (2000) 1-98.
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[PhD Thesis] A. Savas-Halilaj
Title: Complete minimal hypersurfaces with zero Gauss-Kronecker curvature in 4-dimensional space forms
University of Ioannina, Greece, (2006) 1-132.