arXiv subjects

Fritz Hiesmayr

Publications and source records attributed to Fritz Hiesmayr.

3 recordsLinked to original sources

arXiv2020

A Bernstein theorem for two-valued minimal graphs in dimension four

We prove a Bernstein-type theorem for two-valued minimal graphs in the four-dimensional Euclidean space $\mathbf{R}^4$. This states that two-valued functions defined on the entire $\mathbf{R}^3$, and whose graph is a minimal surface, must necessarily be linear. This is a two-valued analogue of the classical Bernstein theorem, which asserts that in dimensions up to $n+1 \leq 8$, an entire single-valued minimal graph is linear. The main contrast with the single-valued theory is the presence of a large set of singularities in the graphs of two-valued functions. Indeed two-valued minimal graphs are neither area-minimising, nor is the regularity theory of elliptic PDE directly available in this setting. We obtain structure results for the blowdown cones of two-valued minimal graphs, valid in dimension $n+1 \leq 7$, proving in particular that they are smoothly immersed away from an $(n-2)$-rectifiable set that includes its branch points. In dimension four we go further, and completely classify the possible blowdown cones using a combinatorial argument. We show that they must be a union of two three-dimensional planes: this is the key to the proof of the Bernstein theorem.

math.DG↗

arXiv2020

Rigidity of low index solutions on $S^3$ via a Frankel theorem for the Allen-Cahn equation

We prove a rigidity theorem in the style of Urbano for the Allen-Cahn equation on the three-sphere: the critical points with Morse index five are symmetric functions that vanish on a Clifford torus. Moreover they realise the fifth width of the min-max spectrum for the Allen-Cahn functional. We approach this problem by analysing the nullity and symmetries of these critical points. We then prove a suitable Frankel-type theorem for their nodal sets, generally valid in manifolds with positive Ricci curvature. This plays a key role in establishing the conclusion, and further allows us to derive ancillary rigidity results in spheres with larger dimension.

math.DG↗

arXiv2018

Spectrum and index of two-sided Allen-Cahn minimal hypersurfaces

The combined work of Guaraco, Hutchinson, Tonegawa and Wickramasekera has recently produced a new proof of the classical theorem that any closed Riemannian manifold of dimension $n + 1 \geq 3$ contains a minimal hypersurface with a singular set of Hausdorff dimension at most $n-7$. This proof avoids the Almgren--Pitts geometric min-max procedure for the area functional that was instrumental in the original proof, and is instead based on a considerably simpler PDE min-max construction of critical points of the Allen--Cahn functional. Here we prove a spectral lower bound for the hypersurfaces arising from this construction. This directly implies an upper bound for the Morse index of the hypersurface in terms of the indices of the critical points, provided it is two-sided. In particular, two-sided hypersurfaces arising from Guaraco's construction have Morse index at most $1$. Finally, we point out by an elementary inductive argument how the regularity of the hypersurface follows from the corresponding result in the stable case.

math.DG↗