Horizontal Curve Shortening Flow

• Horizontal curve shortening flow is a curvature-driven evolution enhanced by incorporating a preferred horizontal structure across various geometric settings.
• It encompasses distinct formulations such as horizontal graphs on warped surfaces, geodesic translations in hyperbolic space, and Legendrian shifts in contact geometry.
• The theory reveals critical insights into uniqueness challenges and long-term convergence, often resulting in solutions aligning with horizontal geodesics or straight lines.

Searching arXiv for recent and relevant papers on “horizontal curve shortening flow” and closely related formulations. Horizontal curve shortening flow is not a single universally fixed equation in the cited literature. Rather, the phrase and closely related constructions occur in several settings in which ordinary curve shortening flow is supplemented by a preferred horizontal structure: a horizontal graph parametrization on a warped product surface, a geodesic translation in a horizontal direction in the hyperbolic plane, a Legendrian curve constrained to the horizontal/contact distribution in $\mathbb R^3$, or a complete-curve formulation in which the tangent is aligned with a fixed horizontal vector $e_1$. In all of these settings, the underlying geometric law remains motion by curvature or geodesic curvature; in the Euclidean plane, for example, curve shortening is the condition $v_n=\kappa$ (McDonald, 2021). What changes is the meaning of “horizontal,” the analytic form of the evolution, and the associated existence, uniqueness, and asymptotic theory (Peachey, 2022, Krznarić et al., 13 May 2026, Drugan et al., 2015, Miura et al., 4 Apr 2025).

1. Terminological scope

Across the cited papers, “horizontal” refers to distinct geometric structures rather than to a single canonical model. One paper states explicitly that it does not use the phrase “horizontal curve shortening flow” as a separate formal theory, while nevertheless developing horizontal/vertical graph versions of curve shortening flow on a warped surface (Peachey, 2022). Other papers use the term for motion relative to a chosen horizontal direction in $\mathbb H^2$, for Legendrian curves tangent to the horizontal distribution $\ker\eta$ in contact $\mathbb R^3$, or for complete curves whose tangent is energetically driven toward a fixed horizontal vector $e_1$ (Krznarić et al., 13 May 2026, Drugan et al., 2015, Miura et al., 4 Apr 2025).

Context	Meaning of “horizontal”	Explicit formulation
Warped metric $g=dx^2+e^{2\phi(x)}dy^2$	horizontal graph $x=x(y,t)$	$\mathcal H(x)=0$ (Peachey, 2022)
Hyperbolic plane $e_1$0	horizontal geodesic translation $e_1$1	$e_1$2 (Krznarić et al., 13 May 2026)
Contact $e_1$3	tangent to $e_1$4	$e_1$5 (Drugan et al., 2015)
Complete curves in $e_1$6	tangent alignment with $e_1$7	$e_1$8 (Miura et al., 4 Apr 2025)

This multiplicity of usages suggests that “horizontal curve shortening flow” is best understood as an umbrella expression for curvature-driven evolutions constrained or organized by a preferred horizontal direction, foliation, or distribution, rather than as a single standard PDE.

2. Horizontal graphs on symmetric and warped surfaces

A precise graphical formulation appears for complete Riemannian surfaces with warped product metric

$e_1$9

If a curve is written as a graph $v_n=\kappa$0, described in the paper as a “horizontal graph,” or as a graph $v_n=\kappa$1, then curve shortening flow becomes a quasilinear parabolic PDE (Peachey, 2022). The operators are

$v_n=\kappa$2

$v_n=\kappa$3

with

$v_n=\kappa$4

In that paper, the horizontal formulation is precisely the PDE $v_n=\kappa$5, and the vertical formulation is $v_n=\kappa$6 (Peachey, 2022).

The same work places this graphical theory inside a global uniqueness problem for proper curves. For uniformly proper solutions, uniqueness is formulated by requiring that two solutions with identical initial data have the same image for as long as both flows exist. The paper records the conjecture that curve shortening flow is unique on the flat plane, but proves that uniqueness can fail for a smooth complete metric $v_n=\kappa$7 on $v_n=\kappa$8 (Peachey, 2022). The mechanism is “blooming at infinity”: for a rotationally symmetric metric, geodesic circles satisfy

$v_n=\kappa$9

and blooming at infinity means there exists a solution with $\mathbb H^2$0 as $\mathbb H^2$1. In the constructed metric, $\mathbb H^2$2 for large $\mathbb H^2$3, and a uniformly proper non-static solution starts from the $\mathbb H^2$4-axis, which is a properly embedded geodesic, but immediately peels away at infinity (Peachey, 2022).

This provides an important caveat for horizontal graph formulations. The local PDE is quasilinear parabolic, but global uniqueness is sensitive to geometry at infinity rather than merely to local regularity. In that sense, horizontal curve shortening flow on noncompact symmetric surfaces is governed as much by ambient asymptotics as by the intrinsic curvature law.

3. Horizontal translation and soliton models

A second major usage appears in the hyperbolic plane $\mathbb H^2$5, where the paper introduces “geodesic translations” by moving points along geodesics and then classifies all curve shortening flows evolving by such translations (Krznarić et al., 13 May 2026). For horizontal geodesic translation, the prescribed direction is

$\mathbb H^2$6

In this case the paper proves that the solution is necessarily an equidistant line and gives the explicit evolution

$\mathbb H^2$7

with each time-slice an equidistant line of curvature

$\mathbb H^2$8

As $\mathbb H^2$9, these curves approach the geodesic $\ker\eta$0 (Krznarić et al., 13 May 2026). For general nonzero $\ker\eta$1 with $\ker\eta$2, the curve must again be an equidistant line, with explicit formula and time-dependent curvature given in the paper (Krznarić et al., 13 May 2026).

In the Euclidean plane, an analogous horizontal specialization appears through the Schwarz-function formulation of curve shortening flow. There the evolving analytic curve is encoded by a Schwarz function $\ker\eta$3 satisfying

$\ker\eta$4

and in a moving frame $\ker\eta$5 translating in the positive real direction with unit speed one writes

$\ker\eta$6

This reduces the PDE to

$\ker\eta$7

whose solution yields the grim reaper

$\ker\eta$8

The paper states explicitly that this is the “horizontal” specialization, namely translation in the positive real direction (McDonald, 2021).

These two models—hyperbolic horizontal geodesic translation and Euclidean horizontal translation in a moving frame—show that “horizontal” often singles out a preferred one-parameter family of ambient motions. In the Euclidean case, the distinguished translator is the grim reaper; in the hyperbolic case, the distinguished translating solutions are rigidly forced into the constant-curvature class, and the horizontal case selects equidistant lines (McDonald, 2021, Krznarić et al., 13 May 2026).

4. Horizontal/contact formulations in $\ker\eta$9

A different meaning of horizontality arises in contact geometry. In $\mathbb R^3$0 with coordinates $\mathbb R^3$1, contact form

$\mathbb R^3$2

metric

$\mathbb R^3$3

and Reeb field

$\mathbb R^3$4

the horizontal distribution is $\mathbb R^3$5 (Drugan et al., 2015). A curve $\mathbb R^3$6 is Legendrian exactly when

$\mathbb R^3$7

equivalently

$\mathbb R^3$8

Legendrian curve shortening flow is defined by

$\mathbb R^3$9

where $e_1$0 is chosen so that the Legendrian condition is preserved (Drugan et al., 2015). More generally, if

$e_1$1

then the paper proves that $e_1$2 stays Legendrian iff

$e_1$3

This is the horizontal/contact compatibility relation (Drugan et al., 2015).

The decisive fact is that the planar projection

$e_1$4

evolves by ordinary planar curve shortening flow whenever $e_1$5 solves Legendrian curve shortening flow (Drugan et al., 2015). The projected equations

$e_1$6

are exactly the planar curvature-flow velocity. Thus the three-dimensional horizontal flow is, in the paper’s formulation, a horizontal lift of planar curve shortening flow. The singularity theory is then transferred from the plane: for balanced figure-eight projections satisfying zero signed area, symmetry about an interior axis, and exactly two inflection points, the planar curve collapses to a point at the first singular time (Drugan et al., 2015).

This contact-geometric interpretation makes “horizontal curve shortening flow” literally distributional: the evolving curve is constrained to remain tangent to a horizontal bundle, but the effective curvature evolution is read off from a two-dimensional projection.

5. Graph preservation, eventual graphicality, and convergence to horizontal geodesics or lines

Several papers study curve shortening flow relative to a preferred horizontal foliation or axis and show that graphicality is either preserved or eventually attained. For warped product manifolds $e_1$7 with metrics

$e_1$8

a curve is called graphical over $e_1$9 if it can be written as

$g=dx^2+e^{2\phi(x)}dy^2$0

The associated angle functions

$g=dx^2+e^{2\phi(x)}dy^2$1

remain positive under curve shortening flow by maximum-principle estimates, and a closed initial graphical curve exists for all time and converges smoothly to a totally geodesic closed curve (Zhou, 2015).

A related eventual-graph result is proved on surfaces

$g=dx^2+e^{2\phi(x)}dy^2$2

with $g=dx^2+e^{2\phi(x)}dy^2$3, which are described as not convex at infinity (Fujihara, 2024). The graph condition is

$g=dx^2+e^{2\phi(x)}dy^2$4

with angle function

$g=dx^2+e^{2\phi(x)}dy^2$5

If the initial embedded curve is not null-homotopic, then the flow exists for all time and $g=dx^2+e^{2\phi(x)}dy^2$6 as $g=dx^2+e^{2\phi(x)}dy^2$7. Under either

$g=dx^2+e^{2\phi(x)}dy^2$8

or the smallness condition

$g=dx^2+e^{2\phi(x)}dy^2$9

there exists $x=x(y,t)$0 such that

$x=x(y,t)$1

so the curve becomes a graph after finite time (Fujihara, 2024).

For complete curves in $x=x(y,t)$2, the horizontal direction is built directly into the energy. The direction energy

$x=x(y,t)$3

is equivalent to

$x=x(y,t)$4

and the paper interprets curve shortening as “tangent aligning” toward the fixed horizontal vector $x=x(y,t)$5 (Miura et al., 4 Apr 2025). If

$x=x(y,t)$6

then for a global solution of

$x=x(y,t)$7

one has

$x=x(y,t)$8

and after arclength reparametrization and translation the limit is the horizontal line

$x=x(y,t)$9

(Miura et al., 4 Apr 2025).

Taken together, these results show that horizontal structure can enter either as an invariant graph condition, as an eventual graph condition, or as an asymptotic tangent-selection principle. The recurrent conclusion is convergence to a geodesic or straight line distinguished by the chosen horizontal direction.

6. Stability, uniqueness, and relation to classical planar theory

The analytical behavior of horizontal formulations remains closely tied to the classical planar theory of curve shortening flow. For planar Jordan curves of zero Lebesgue measure, the evolution by curvature flow or its weak level-set formulation depends continuously on the initial curve in Fréchet distance. The central continuity statement is

$\mathcal H(x)=0$0

and for such zero-area Jordan curves the level-set flow immediately becomes a smooth Jordan curve and coincides with the classical curvature flow for $\mathcal H(x)=0$1 (Ma, 2023). This gives a rigorous stability theorem for shape-preserving perturbations of closed planar data.

At the same time, horizontal or graphical settings expose new uniqueness issues. On certain symmetric warped planes, the same properly embedded geodesic can support both the static solution and a non-static uniformly proper solution, so uniqueness fails (Peachey, 2022). The contrast with the planar Fréchet-continuity theorem is instructive: continuity for closed Jordan curves and nonuniqueness for proper noncompact curves are both consequences of the interaction between curvature flow and the topology imposed on the initial class, but they occur in sharply different ambient settings (Ma, 2023, Peachey, 2022).

Explicit model solutions remain central throughout. In the Schwarz-function description of planar curve shortening flow, the circle, grim reaper, paperclip, and hairclip are all recovered from the equation

$\mathcal H(x)=0$2

and the grim reaper is the canonical steady horizontal translator in the moving frame $\mathcal H(x)=0$3 (McDonald, 2021). In hyperbolic geometry, the corresponding horizontal-geodesic-translation solitons are equidistant lines rather than grim-reaper curves (Krznarić et al., 13 May 2026). In contact geometry, the three-dimensional horizontal problem is reduced to planar curve shortening by projection (Drugan et al., 2015). In the energy-based theory of complete curves, the distinguished asymptotic object is the horizontal line $\mathcal H(x)=0$4 (Miura et al., 4 Apr 2025).

These developments indicate that horizontal curve shortening flow is not a replacement for classical curve shortening flow but a family of refinements of it. The common invariant is the curvature law; the distinctive content lies in how a horizontal direction, graph structure, or horizontal distribution reorganizes existence theory, comparison principles, uniqueness, and asymptotic classification.