The nucleus in many cell types is a stiff organelle, and yet fat-filled lipid droplets (FD's) in the cytoplasm can indent and displace the nucleus. FD's are phase-separated liquids with a poorly understood interfacial tension that determines how FD's interact with other organelles. Here, micron-sized FD's remain spherical while deforming both the nuclear lamina and peri-nuclear actomyosin. Nuclear lamins are intermediate filaments with a persistence length in the range of ∼0.2-1.5 μm with Lamin-B1 directly attached to the inner nuclear membrane via farnesylation groups that are lacking in mature lamin-A,C. Such rigidity can compromise the attachment of Lamin-B1 to the membrane in cases of high Gaussian curvature as seen through local dilution of Lamin-B1 at sites of indentation which is independent of Lamin-A,C. A single filament model of curvature-driven membrane detachment fits to Lamin-B1's relative density with the association energy being the key parameter to fit the experimental data. Lamin-B1 depletion triggers rupture as indicated by persistent, local accumulation of cytosolic DNA sensor cGAS at the nuclear boundary. FD-nucleus interactions also initiate rapid mis-localization of the DNA repair factor KU80, confirming nuclear rupture, and associate with heightened DNA damage and perturbed cell cycle. Similar results are evident in FD-laden cells after constricted 3D-migration, which is impeded by FD's. Spherical shapes of small FD's are consistent with a high interfacial tension that we measure for FD's mechanically isolated from fresh adipose tissue as ∼40 mN/m - which is far higher than other liquid condensates, but typical of oils in water and sufficiently rigid to disrupt cell structures.