Paper proves unique tangent maps for complex maps into algebraic varieties.
problem Proving uniqueness of tangent maps for complex maps into algebraic varieties.
method Developed techniques to prove uniqueness of tangent maps for weakly holomorphic and locally approximable maps.
result Established unique tangent cone property for weakly holomorphic maps into projective algebraic varieties.
Uniqueness proven for cylindrical tangent cones in high dimensions.
problem Proving uniqueness of cylindrical tangent cones in high dimensions.
method Analyzing area-minimizing hypersurfaces in R^9.
result Uniqueness of cylindrical tangent cones Cp,qimesR in R9. We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent …
Uniqueness proven for stable hypersurface tangent cones.
problem Stability and uniqueness of tangent cones for stable hypersurfaces.
method Analysis of isolated singularities and tangent cones of stable minimal hypersurfaces.
result Uniqueness of tangent cones with integer multiplicities.
Analytic sets with unique infinite tangent cone are algebraic.
problem Characterizing analytic sets with unique infinite tangent cones.
method Analytic and algebraic set properties, degree of complex algebraic sets.
result Degree of Lipschitz normally embedded sets equals their infinite tangent cone degree.
Paper proves unique conical tangent flows in all dimensions.
problem Proving uniqueness of conical singularities in mean curvature flows.
method Based on Chodosh-Schulze's work in hypersurfaces, extended to all codimensions.
result Uniqueness of asymptotically conical tangent flows in all codimensions.
Study shows unique tangent cones for area-minimizing currents at boundary points.
problem Uniqueness of tangent cones for area-minimizing currents with arbitrary multiplicity.
method Analysis of area minimizing currents in C2 submanifolds with arbitrary boundary multiplicity. result Tangent cones are unique at density Q/2 boundary points. Unique cylindrical tangent cone for Simons' hypersurface found.
problem Uniqueness of cylindrical tangent cones for area-minimizing hypersurfaces.
method Developed a new Lojasiewicz inequality for non-isolated singularities.
result Cylindrical tangent cone for Simons' hypersurface is unique.
Unique tangent cones found for Kahler-Einstein metrics on singular varieties.
problem Finding unique tangent cones for Kahler-Einstein metrics on singular varieties.
method Analyzing unique Ricci flat currents with local bounded potential.
result Local tangent cones of the unique Kahler-Einstein metric are unique.
Study proves uniqueness of Yang-Mills field tangent cones in arbitrary dimensions.
problem Proving uniqueness of Yang-Mills field tangent cones.
method Log-epiperimetric inequality, Luckhaus type lemma, and curvature concentration exclusion.
result Uniqueness of tangent cones for Yang-Mills fields in arbitrary dimensions.
Researchers prove uniqueness of certain cylindrical tangent cones for special Lagrangians.
problem Proving uniqueness of cylindrical tangent cones for special Lagrangians.
method Analyzing exact special Lagrangian submanifolds with multiplicity one and cylindrical tangent cones.
result The cylindrical tangent cones are unique under specific conditions.
We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex man…
This part II of the paper is concerned with questions of existence and uniqueness of tangents in the special case of G-plurisubharmonic functions, where G is a compact subset of the Grassmannian of p-planes in Rn. An upper semi-continuous function u on an open set Ω in Rn is G-plurisubharmon…
New proof of harmonic map uniqueness with analytic targets.
problem Uniqueness of energy-minimizing harmonic maps with analytic targets.
method Symmetric (log)-epiperimetric inequality for harmonic maps with analytic targets.
result Tangents at infinity of energy-minimizing harmonic maps are unique.
Study on singularities in Lagrangian mean curvature flow with special Lagrangian cones.
problem Understanding singularities in Lagrangian mean curvature flow.
method Analysis of tangent flows and blowup limits of special Lagrangian cones.
result Uniqueness of tangent flows in dimension two and any dimension when the link is connected.
Study proves uniqueness of tangent cones for area-minimizing currents in higher codimensions.
problem Understanding the fine structure of singular points in area-minimizing currents.
method Analysis of tangent cones and application of previous work.
result Uniqueness of tangent cones at Hm−2-a.e. points in the support of area-minimizing currents. Paper proves strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
problem Proving strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
method Established Lojasiewicz inequality for pointed W-entropy under cylindrical geometry assumption. result Strong uniqueness of cylindrical tangent flows at first singular time of Ricci flow proved.
Proves uniqueness of cylindrical tangent flows in mean curvature flow.
problem Proving uniqueness of cylindrical singularity models in mean curvature flow.
method Inspired by Székelyhidi's approach, uses a different method to prove uniqueness.
result Proves uniqueness of cylindrical tangent flows.
In this note, we combine the work of Ilmanen and of Colding-Ilmanen-Minicozzi to observe a uniqueness property for tangent flows at the first singular time of a smooth mean curvature flow of a closed surface in 3-dimensional Euclidean space. Specifically, if, at a fixed singular point, one tangent flow is a positive in…
There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation f(D2u)=0. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman …
Study on singularities in area-minimizing currents, proving unique tangent cones and rectifiability.
problem Understanding singularities in area-minimizing currents.
method Fine excess decay theorems and almost monotonicity of a frequency function.
result Unique tangent cones and countably (m−2)-rectifiable singular set. The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.
problem Characterizing and understanding Ricci flows with closed and smooth tangent flows.
method Analyzing ancient and finite-time singularity Ricci flows to prove uniqueness and characterizations.
result The tangent flow is unique and characterizes ancient and finite-time singularity flows.
It is known that for each combinatorial type of convex 3-dimensional polyhedra, there is a representative with edges tangent to the unit sphere. This representative is unique up to projective transformations that fix the unit sphere. We show that there is a unique representative (up to congruence) with edges tangent to…
Study shows unique tangent flow for Lagrangian surfaces with bounded mean curvature.
problem Understanding the behavior of Lagrangian surfaces with bounded mean curvature.
method Analyzing zero Maslov Lagrangian mean curvature flow in C2 with bounded mean curvature. result The tangent flow at a singular point is unique if the mean curvature stays uniformly bounded.
We prove that tangent cones at singular boundary points of a two-dimensional current almost area minimizing are unique. Following the ideas exposed by White in [8], the result is achieved by combining a suitable epiperimetric inequality and an almost-monotonicity formula for the mass at boundary points.
Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying all invariant contact metric structures on tangent sphere bundles of compact rank-one symmetric spaces.
method Explicitly obtained all structures, distinguishing K-contact, Sasakian, and 3-Sasakian structures.
result There is a unique Sasakian-Einstein metric on tangent sphere bundles of spheres and real projective spaces.
Paper proves unique tangent flow at infinity for entropy-limited curve shortening.
problem Proving uniqueness of tangent flows for finite-entropy curve shortening.
method Rescaled backward convergence to a line, entropy analysis, and geometric properties.
result Ancient smooth curve shortening flow has a unique tangent flow at infinity.
Paper constructs infinitely many tangent functors on diffeological spaces.
problem Tangent spaces in diffeological spaces are not uniquely defined.
method Introduced and constructed infinitely many non-isomorphic tangent functors.
result The choice of tangent functor is not unique outside smooth manifolds.
Proves rectifiability for specific metric spaces with unique tangents.
problem Rectifiability of CD(K,N) and MCP(K,N) spaces with unique tangents. method Failure of CD condition in sub-Finsler Carnot groups, new result on MCP spaces, recent breakthrough by Bate. result Proves rectifiability for CD(K,N) and MCP(K,N) spaces under specific conditions. New height estimate for area minimizing currents, leading to unique tangent cones and decay properties.
problem Analyzing singularities of area minimizing currents.
method Height estimate, decay estimates, techniques inspired by previous works.
result Locally area minimizing currents have a unique tangent cone at almost every point and decay rapidly to a unique tangent plane at branch points.
We consider 2-dimensional integer rectifiable currents which are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by a suitable modification of White's original theorem for area minimizing currents…
A theorem proves integrability of Fréchet tangent distributions.
problem Integrability of Fréchet tangent distributions on manifolds.
method Introduced Condition W, applied variational approach, used differential forms.
result Existence and uniqueness of maximal foliations.
Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In …
The paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
problem Proving strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
method Establishing a Lojasiewicz inequality for the pointed W-entropy in Ricci flow under the assumption of geometry near the base point being close to a generalized cylinder. result Proves strong uniqueness of generalized cylindrical tangent flows and shows that the subset of points with rectifiable Sqck(N) is horizontally parabolic. We proved a uniqueness theorem of tangent connections for a Yang-Mills connection with an isolated singularity with a quadratic growth of the curvature at the singularity. We also obtained controls over the rate of the asymptotic convergence of the connection to the tangent connection under assumptions that the connect…
We consider the Calabi-Yau metrics on Cn constructed recently by Yang Li, Conlon-Rochon, and the author, that have tangent cone C×A1 at infinity for the (n−1)-dimensional Stenzel cone A1. We show that up to scaling and isometry this Calabi-Yau metric on Cn is unique. We al…
We consider positive-(1,1) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood. Without this assumption, counterexamples to the uniqueness of tangent cones can be prod…
We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on μ-polystable holomorphic bundles over Pn−1.
Proves uniqueness of blowups for forced mean curvature flow.
problem Proving uniqueness of blowups for forced mean curvature flow.
method Adapting methods from Euclidean space mean curvature flow to handle forcing term and blow-up limits.
result Uniqueness of tangent cones for forced mean curvature flow at self-shrinkers and cylindrical self-shrinkers.
Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.
problem Constructing minimal hypersurfaces with specific geometric properties.
method Constructing minimal hypersurfaces with cylindrical tangent cones and proving unique continuation results.
result Existence and properties of minimal hypersurfaces with cylindrical tangent cones.
The study examines singularities in flows with curvature bounds and identifies unique tangent flows.
problem Analyzing singularities in mean curvature flows with curvature bounds.
method Examines tangent flows and uses stationary and area-minimizing cones to identify unique flows.
result For flows with H∈L∞Llocp, the tangent flow is unique when p=∞ and C is a regular cone. This short note has been written as an Oberwolfach report for the workshop "Differentialgeometrie im Grossen". We discuss properties of metric spaces that at almost all points admit a tangent metric space. We explain why, under some mild assumptions, the tangents are almost surely subFinsler Carnot groups. We mention s…
Study area minimizing currents in Riemannian manifolds, proving unique structure and decay.
problem Area minimizing currents in Riemannian manifolds with moduli.
method Structural results and uniqueness theorem, inspired by Simon's techniques.
result Uniqueness and decay towards tangent cones for area minimizing currents.
We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the …
We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapion…
We show, for mean curvature flows in Euclidean space, that if one of the tangent flows at a given space-time point consists of a closed, multiplicity-one, smoothly embedded self-similar shrinker, then it is the unique tangent flow at that point. That is the limit of the parabolic rescalings does not depend on the chose…
It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume…
The article calculates the F-convergence rate for Ricci flows with closed and smooth tangent flows.
problem Analyzing the convergence rate of Ricci flows with specific tangent flows.
method Calculating the F-convergence rate for Ricci flows with closed and smooth tangent flows. result A Ricci flow with closed and smooth tangent flow is ∣logλ∣−θ close to its tangent flow in the F-sense.